Shouldn't Mary’s genealogy be Jesus' family history?

Jesus from both biological and scriptural perspective is the Son of Mary. He is a descendant of of David through Mary and not Joseph. Matthew's account suggests that there are 14 generations from Captivity in Babylon until Jesus Christ but this does not add-up. Count for yourself. The only correct account is Luke's which captures Mary's genealogy but was mistranliterated (not by the writer but the translators having been sinceerely influenced by the idea that Jesus must come from Royal line of King David) as Joseph's. It was predicted from the Begining that the seed of the Woman would bruise the head of the Serpent. With Mary, there are a complete 14 generations from the Captivity in Babylon.

One thing that I found disturbing in comments for this question is to read that God was the father of Jesus. For Christians, it is blasphemous to say that God had a son. Jesus being called the Son of God was to portray that Jesus was same in likeness as God which meant that He was God incarnate.

Regarding lineage of Jesus it just a notional that Jesus was assigned to an earthly father and mother. Jesus was prophesied to come to a family with a lineage link to king David. Gospel of Mathew gives us the lineage of Jesus from David towards Joseph and Luke gives the lineage of Jesus from David towards Mary.

Mathew is indicating that Joseph is from David through Solomon and Luke is indicating that Mary is also from David through Nathan. That is why we see differences in the lineages given by Luke and Mathew. Luke’s lineage is thru Jesus maternal grandfather Eli.

Luke’s text says that Jesus was “a son, as was supposed, of Joseph, of Eli”. This been understood as acknowledgment of the virgin birth. The parenthetical expression: “a son (as was supposed of Joseph) of Eli” indicate that Jesus is called a son of Eli because Eli was his maternal grandfather, his nearest male ancestor on His mother side. A variation on this idea is to explain “Joseph son of Eli” as meaning a son-in-law, perhaps even an adoptive heir to Eli through his only daughter Mary. An example of the Old Testament use of such an expression is Jair, who is called “Jair son of Manasseh” but was actually son of Manasseh’s granddaughter.

Jesus being an Eternal God chose a human instrument to come into this world in a miraculous way, he chose the path as the prophecies predicted his human lineage. So, Jesus getting into a family by a miracle as well as his human lineage, go hand in hand. There is no contradiction, but each complement and give a whole picture for our understanding.

I think it was for the same reason that Jesus was Baptized and the same reason Mary was purified.

 Jesus replied, “Let it be so now; it is proper for us to do this to fulfill all righteousness.” Then John consented. (Matt 3:15 NIV)

 When the time came for the purification rites required by the Law of Moses, Joseph and Mary took him to Jerusalem to present him to the Lord   (Luke 2 22-23)

Jesus' recorded genealogies go through His foster father Joseph to "fulfill all righteousness" and solidify him as a son of kings and of prophets under the law. Mary was purified, even though she was without sin and knew no man.

It should be noted, that Mary, as tradition has it, was also a daughter in the line of David and probably a not terribly distant relative of her husband. So, for the most part, the genealogy is Mary's as well.

Just because someone is not literally born of a specific man, does not discount him from that lineage. Joseph was not the cause of Jesus' birth, but he certainly accepted Him into his family and helped raise him well.

We see this in the story of Abraham, where even his servants under his household were instructed to receive the symbol of circumcision, and thus were considered to be from Abraham's line.

So Abraham took Ishmael his son, all who were born in his house and all who were bought with his money, every male among the men of Abraham’s house, and circumcised the flesh of their foreskins that very same day, as God had said to him.

That very same day Abraham was circumcised, and his son Ishmael;27 and all the men of his house, born in the house or bought with money from a foreigner, were circumcised with him.

From that Day forward they were all considered to be descendants of Abraham.

You have to understand that the purpose of the genealogy is literary, and not scientific. The Gospel of Matthew has a primary focus on Jesus as a great and authoritative teacher (like Moses)—notice how the genealogy starts at Abraham, the father of the Israelites. By contrast, Luke's genealogy (Lk 3:23ff.) traces Jesus' lineage back to Adam, which many have speculated is meant to illustrate the universality of Jesus prophetic message of liberation found in Luke–Acts.

In other words, since the purpose of the genealogies is not scientific, which parent the lineage goes through doesn't matter; but the father is clearly preferred.

The lineage in Matthew follows Joseph, which would be Jesus' legal lineage to David.

The lineage in Luke follows Mary, which would be Jesus' natural lineage to Adam.

Both run through David, which qualifies him as an heir by either natural or legal lineage.