Shouldn't Mary’s genealogy be Jesus' family history? Matthew 1:1-17 provides a genealogy of Jesus, traced back to Abraham:


  The Family History of Jesus the Messiah:
  .  .  .
  Abraham was the father of Isaac
  .  .  .
  Matthan was the father of Jacob.
  Jacob was the father of Joseph.
  Joseph was the husband of Mary,
  and Mary was the mother of Jesus, who is called the Messiah. (from Matthew 1:1-2, 15-16, ERV) 


However, next it says that Joseph had nothing to do with the birth of Jesus: that Jesus was born of the seed of the Holy Spirit, not Joseph's seed:


  This is how the birth of Jesus the Messiah happened. His mother Mary was engaged to marry Joseph. But before they married, he learned that she was expecting a baby. (She was pregnant by the power of the Holy Spirit.) (Matthew 1:18, ERV)


And:


  When Joseph woke up, he did what the Lord's angel told him to do. He married Mary. But Joseph did not have sexual relations with her until her son was born. And he named him Jesus. (Matthew 1:24-25, ERV)


How is Matthew 1:1-17 the family history of Jesus?

As far as any family lines were involved in Jesus' birth as a human, shouldn't Mary's genealogy be Jesus' family history? (I truly don't get this.)
There are some debates about this because these genealogies do not have every name along the branches but certain representative names. Most sources I have encountered think Matthew proves Christ was legally entitled to the throne of David as a legal ancestor to David through his father by law.  Luke on the other hand traces the  the physical lineage through Mary, who also was a descendent of David.

In any case as far as I can tell, although we have lost all the genealogical records when Jerusalem was destroyed, these records did not seem to cause any controversy among the early christians but effectively added further proof of the legitimate claims that Jesus was the Messiah.  To those who knew about the genealogies at the time this was important. For us we can only take that 'battle won' as historical proof.  Furthermore, with these records now lost no new Messiah can claim to have the correct genealogical record. Jesus was the Messiah, or there never was such a thing - case closed.