Inheritance is indeed through the father. Messianic Jewish scholar David Stern notes in his Jewish New Testament Commentary:

    "Yosef's behavior shows that he accepted Yeshua as his son. According to the Mishna, "If one say, 'This is my son, 'he is to be believed" (Bava Batra 8:6). The Gemara explains that he is believed "as regards the right of inheritance" (Bava Batra 134a). Thus Yeshua, as legally acknowledged son, is entitled to inherit the throne of King David from Yosef, a descendant of David (v. 8). (This point is made by Phillip Goble, How to Point to Yeshua in Your Rabbi's Bible, New York: Artists for Israel, 1986.)

Glenn Miller, of the Christian-ThinkTank notes:

    "There are TONS of references to Jesus as being in the lineage of David, throughout the NT, and NOT just in the genealogies.

For example:

In Zechariah's Song - Luke 1:69
The blind man at Jericho - Mt 9:27; Mr 10:47
The Canaanite Woman (a foreigner!) - Mt 15:22
The questioning crowd in Mt 12:23
The massive crowd at the Triumphal Entry - Mt 21:15
Apostle Peter - Acts 2.25ff
Apostle Paul - Acts 13.22ff; Romans 1.3; 2 Tim 2.8
Apostle John - Revelation 5:5; 22.16"

Messiah descends from Jehoiakhin
Tanhuma Genesis, Toledot (8th-9th c.)

Scripture alludes here to the verse Who art thou, O great mountain before Zerubbabel? Thou shalt become a plain (Zech. 4:7). This verse refers to the Messiah, the descendant of David. . . .From whom will the Messiah descend? From Zerubbabel.
--Midrash Tanhuma-Yelammedenu, translated by Samuel A. Berman (Hoboken, NJ: Ktav, 1996), p. 182.

   Malbim calls to our attention that in the prophecy of Haggai (2:23), God says, "On that day I will take you, Zerubbabel, and I will make you like a signet," for the King Messiah will be like a signet ring on God's right hand, so to speak. Just as the name of the owner of the ring is engraved on his signet ring, through which he makes himself known, so will God's name be known in the world through the King Messiah, through whom His miracles will be known. He says here that, though, in the future, Coniah will be the signet on My right hand, for the Messiah will spring from his seed, now I will remove him from there.
--Ibid., p. 183. Malbim is an acronym for Meir Loeb ben Jehiel Michale, a 19th c. rabbi and commentator. 22:24.

Sanhedrin 37a, Soncino Talmud

R. Johanan said: Exile atones for everything, for it is written, "Thus saith the Lord, write ye this man childless, a man that shall not prosper in his days, for no man of his seed shall prosper sitting upon the throne of David and ruling any more in Judah." ^a Whereas after he [the king] was exiled, it is written, And the sons of Jechoniah,-the same is Assir-Shealtiel his son etc.^b [He was called] Assir,^c because his mother conceived him in prison. Shealtiel, ^d because God did not plant him in the way that others are planted. . . Another interpretation: Shealtiel, because God obtained ^e [of the Heavenly Court] absolution from His oath."^f

footnotes:
a. Jer. XXII, 30  .
b. I Ch. II, 17. Notwithstanding the curse that he should be childless and not prosper, after being exiled he was forgiven.
c. ASYR, imprisoned.
d. AL ShTLV, a play on ShALTYAL.
e. ShAL AL, 'God asked."
f. Which He had made, to punish Jechoniah with childlessness.

R. Shabbethai said: He [Jeconiah] did not move thence before the Holy One, blessed be He, pardoned him all his sins. Referring to this occasion Scripture has said: Thou art all fair, my love, and there is no blemish in thee (S.S. IV, 7). A Heavenly Voice went forth and said to them: 'Return, ye backsliding children, I will heal your backslidings' (Jer. III, 22).
--Soncino Midrash Rabbah vol. 4, p. 249

Why does Matthew delete names from his list?! And isn't that deceitful?!

A. Why does Matthew delete names from the text?
    
    Nazarene scholar, James Trimm, observes in "A Kabbalistic look at Mt. 1:1-18":

 Hidden in the genealogy:

1.  Yeshua the son of DAVID (DVD = 14) then three sets of 14 generations are listed (Mt. 1:18) showing on a SOD level that Yeshua is the son of 14/David.

2.  There are 3 sets of 14 generations (1:18) because 3 * 14 = 42 showing us on a SOD level that Messiah is the son of ELOAH (ELOAH = 42).

3.  The genealogy runs Avraham>David>Messiah thus showing us that he is ADAM KADMON
(the Notarikon of Avraham, David, Messiah is ADAM)

The Hebrew text of Mattityahu 1:1 begins with ALEF and ends with MEM
these two letters keep showing up in important pairs:

Aharon & Moshe

Ester & Mordecai

Eliyahu & Moshe (the two witnesses of Rev. 11?)

The middle letter is BEIT and stands for the word BEN (son). The Son spoke of is the middle pillar of the Godhead.

The First and last letter spell EMA (Mother) and the first and middle letter spell AV (Father) thus showing that the BEN (Son) is the combination of the EMA and the AV and is the middle pillar of the EMA and the AV.

 

1 Chronicles  6	Ezra 7
Zerahiah Meraioth Amariah Ahitub Zadok Ahimaaz Azariah Amariah	Zerahiah Meraioth ------------ ------------ ------------ ------------ ------------ Amariah

Let's look at the differences between the two genealogies:

1.Matt uses 41 names; Luke uses 71!
2.Matt has a VERY specific structure (3 sets of 14 names); Luke's is a simple list
3.Matt has four women (most foreigners); Luke has none
4.Matt's order descends; Luke ascends.
5.Matt starts with Abraham; Luke ends at Adam.

The main difference between the two is that Matt's has a rhetorical/pedagogical structure to it. In other words, it was designed for memory-retention (common practice in his day -- cf. Keener, Bible Background Commentary--NT loc. cit.). The omissions are simply to make the list easier to learn and/or memorize.

Matthew has a fondness for 'threes'. He has three temptations, illustrations of righteousness, miracles of healing, "fear not"s, questions, prayers in Gethsemane, among others. And the "14" in the "3x14" structure of the genealogy is typically attributed to the rabbinic usage of gematria--usage of letters for numbers. In this case, the name "David" in Hebrew has a number-count of 14 (fitting for a section on the Son of David).

His word choice for 'begat' simply means 'progenitor' and allows considerable gaps to exist WITHOUT it being an inaccuracy. (E.g. my great-great-great-grandfather 'begat' me, in Matt's word-choice.)

What this means is that 'omissions' in Matthew are NOT 'problems' at all.

If Luke's Genealogy is Miriam's, then why is Yosef's name there?
Arthur Fruchtenbaum explains,

In his genealogy, Matthew breaks with Jewish tradition and custom. He mentions the names of four women: Tamar, Rahab, Ruth and Bathsheba (who is the one to whom the pronoun "her" in verse six refers). It was contrary to Jewish practice to name women in a genealogy. The Talmud states, "A mother's family is not to be called a family."

Unlike Matthew, Luke follows strict Jewish procedure and custom in that he omits no names and mentions no women. However, if by Jewish custom one could not mention the name of a woman, but wished to trace her line, how would one do so? He would use the name of her husband. (Possible Old Testament precedents for this practice are Ezra 2:61 and Nehemiah 7:63.) (Emphasis mine.)

"This theory shows us in what way Christ was the "Son of David." If Mary was the daughter of Heli, then Jesus was strictly a descendant of David, not only legally, through his reputed father, but actually, by direct personal descent, through his mother. . . This theory affords a very simple explanation of the whole matter. Mary, since she had no brothers, was an heiress; therefore her husband, according to Jewish law, was reckoned among her father's family as his son. So that Joseph was the actual son of Jacob, and the legal son of Heli. In a word, Matthew sets forth Jesus' right to the theocratic crown; Luke, his natural pedigree. The latter employs Joseph's name, instead of Mary's, in accordance with the Israelite law that "genealogies must be reckoned by the fathers, not mothers."
Alleged Discrepancies of the Bible, John W. Haley, pg. 325-326

Glenn Miller another explanation,

The Jewish folk had numerous provisions for cases of inheritance-transfer in extreme cases. One of the more frequent situations that had to be covered (in a land-based, clan-ownership system) was that of childless marriages, or in some cases, of son-less marriages.

One of the more concise statements of how this would apply here, is by J. Stafford Wright in Dict. of New Test. Theol., III. 662:

"Mary's father (Heli?) had two daughters, Mary and the unnamed wife of Zebedee (John 19:25; Matt 27:56). If there were no sons, Joseph would become son of Heli on his marriage,  to preserve the family name and inheritance (cf. Num 27:1-11; 36:1-12, esp. v. 8, which accounts for Mary marrying a man of the family of David.)"

[The main passages in the OT that refer to these various laws are Num 7:1-11; Num 36:1-12; Lev 25:25; Dt 25:5-10. These practices were widespread in the Ancient Near East, and a good discussion of the details in Israel and differences from the ANE can be found in Roland de Vaux, Ancient Israel: Vol 1--Social Institutions. Two famous cases, for good or ill, of these practices are in the story of Ruth (Book of Ruth) and in the story of Tamar (Gen 38:6ff).]

What this 'nets out to' is that Joseph 'married into' Mary's gene-pool...and hence, the virgin birth doesn't stop the lineage "transfer".

In other words, the the physical-gene did NOT come FROM JOSEPH was IRRELEVANT in this case.  Legal standing was related to EITHER 'genes' OR to 'marriage'. (Although it should be pointed out that levirate arrangements like this required close kinship already, and hence, quite a number of overlapping genes).

    A literal translation of the Greek text starting a v. 23 would be "And Yeshua himself was beginning about thirty years, being son, as was supposed, of Yosef, of the Eli, of the Mattat of the L'vi" etc. The questions raised here are: What does it mean to be "of" someone? And which person is being described  as being "of the Eli"?-Yosef or Yeshua?

" . . .Yeshua is "of the Eli" in the sense of being his grandson; while Yeshua's relationship with Yosef is portrayed in the words, "son, as was supposed"-implying not actually;

". . . Luke's language also distinguishes Yosef from Yeshua's direct ancestors by not including the word "the" before "Yosef" in the original Greek. "By the omission of the article, Joseph's name is separated from the genealogical chain and accorded to a place of its own" (F. Rienecker, Praktishces Handkommentar Zu Lukas Evangelium) 1930, p. 302, as cited in A Jewish Christian Response by the Messianic Jew Louis Goldberg)."

Someone reading the original would understand by the missing definite article from Joseph's name that this was not really Joseph's genealogy, but his wife Miriam's.

Furthermore, although many translations of Luke 3:23 read: "...being supposedly the son of Joseph, the son of Eli...," because of the missing Greek definite article before the name of Joseph, that same verse could be translated as follows: "Being the son (as was supposed) of Joseph the son of Heli...".1 In other words, the final parenthesis could be expanded so that the verse reads that although Y'shua was "supposed" or assumed to be the descendant of Joseph, he was really the descendant of Heli. Heli was the father of Miriam. The absence of Miriam's name is quite in keeping with the Jewish practices on genealogies.

How can Zerubabbel and Sh'alti'el be in BOTH genealogies? That proves Yeshua is physically from Jehoiakhin!

Glenn Miller explains,

Now the issue about Shealtiel and Zerubbabel I find intriguing. The argument [made] here is that THEY are descendants of the 'bad Jeconiah' and THEY show up in BOTH the legal AND the physical lineage's of Jesus. And, if the prophecy in Jeremiah is taken to mean a long-range restriction (which I do NOT believe is the case, see above), then we clearly have a problem in the Lukan, physical/gene-stream lineage of Jesus.

But let me ask an impertinent question here. Why do we believe the Shealtiel and Zerubbabel of the two lineage's are THE SAME PEOPLE? Think about it:

They have different parents
They have different children.
They are descended from different sons of David.
Their chronological placements on a time line could differ by as much as a CENTURY! (depending on how the omissions in Matthew are accounted for, and on what the average age of childbearing was.)

THE ONLY THING THEY HAVE IN COMMON ARE THEIR NAMES!

This can hardly be a strong argument for their identity:
1. Zerubbabel was a common name from the early Persian period (539-331bc.), As shown by cuneiform inscriptions from Babylon (see ZPEB , V. 1057)

2.The genealogies themselves have numerous names that repeat WITHIN the genealogy (e.g. Joseph, Mattathias, Judah) without being the same individuals; These names could also be common names.

3.The names in the genealogies are standard, common, everyday names. We have NUMEROUS people named Levi, Amos, Nahum, etc. in the OT accounts. There is just NO REASON to associate the S+Z of Luke with the S+Z of Matthew. (And even the pattern of S-followed-by-Z doesn't carry much weight--families often honored prominent people this way.)

What this means is that the S+Z of Matthew are the S+Z of Jeremiah, and that the S+Z of Luke (whose genes DO reach to Jesus) are a different set, descended from Nathan and not through Solomon-thru-Jeconiah.

Conclusion
    Matthew's genealogy is that of Yosef's (descendant of Shlomo), who would LEGALLY pass the throne to Yeshua, and Luke's Genealogy is that of Miriam's (descendant of Natan), which shows that Yeshua is PHYSICALLY the descendant of David.