Paul, a servant of Jesus Christ, called to be an apostle, separated unto the gospel of God,
(Which he had promised afore by his prophets in the holy scriptures,)
Concerning his Son Jesus Christ our Lord, which was made of the seed of David according to the flesh;
Jesus had to be of the house of David "according to the flesh", which could only be via his mother Mary since Joseph was not involved in the conception of Jesus. But the genealogies in Mat 1:1-16 and in Luke 3.23-38 only mention Joseph; also, Elizabeth is mentioned as a relative of Mary, but she was of the house of Aaron and not a descendant of David. (Luke 1,5 and 36). Since being a descendant of the house and seed of David was necessary, and since Mary did not meet these pre-requisites, Jesus actually could not really have been the promised Messiah, the Christ.
The above argumentation about the physical descent of Jesus does on the one hand take into consideration the biblical record in Rom 1:3, but then only considers assumed tradition rather than knowledge gained from Scripture regarding the statements in Matthew 1 and Luke 3. Only from this arises the suspicion at the end that Jesus perhaps could not even have been the Messiah.
Jesus needed to be, as Romans 1:3 confirms, of the seed of David according to the flesh. "According to the flesh" clearly refers to his physical descend in which from the human side indeed only Mary had part, because she became pregnant by God's working ("of the holy spirit" -- Mat 1:18). This makes it necessary that Mary needed to be of the house and seed of David, and that is exactly what the genealogy given in Matthew 1:1-16 shows, as I have already pointed out in a previous study with the Title Who is the father of Joseph?. Mary was of the house of David, and she was of the royal lineage through David's son Solomon, which gave Jesus based on his descend "according to the flesh" the full rights to the throne of his father David (Mat 1:6). Joseph, the husband of Mary, was also of the house of David, but of a different line through David's son Nathan (Luk 3:31).
The point that Elizabeth, the wife of Zacharias and mother of John the Baptist, was a relative of Mary even though she was of the house of Aaron (Luke 1:36) means simply that one of the ancestors of Mary on her mother's side (!) were of the tribe of Levi. Mary's ancestors on her father's side are of the royal line of the house of David, as the record in Mat 1 shows.
Once again we can see how accurate the revelation in the Scripture is. Jesus Christ is the son of Abraham and also the son of David, and when he returns he will reign on the throne of his father David, as the angel Gabriel already had proclaimed to his mother Mary even before he was conceived (Luke 1:31-33).