Is there any scripture that supports the suggestion that Jesus's mother Mary was a descendant of King David?
In the topic "Genealogy of Jesus Christ" in the Insight on the Scriptures, the subheading Why do the genealogies of Jesus Christ as given by Matthew and by Luke differ? clarifies what the OP asks:
The difference in nearly all the names in Luke’s genealogy of Jesus as compared with Matthew’s is quickly resolved in the fact that Luke traced the line through David’s son Nathan, instead of Solomon as did Matthew. (Lu 3:31; Mt 1:6, 7) Luke evidently follows the ancestry of Mary, thus showing Jesus’ natural descent from David, while Matthew shows Jesus’ legal right to the throne of David by descent from Solomon through Joseph, who was legally Jesus’ father. Both Matthew and Luke signify that Joseph was not Jesus’ actual father but only his adoptive father, giving him legal right. Matthew departs from the style used throughout his genealogy when he comes to Jesus, saying: “Jacob became father to Joseph the husband of Mary, of whom Jesus was born, who is called Christ.” (Mt 1:16) Notice that he does not say ‘Joseph became father to Jesus’ but that he was “the husband of Mary, of whom Jesus was born.” Luke is even more pointed when, after showing earlier that Jesus was actually the Son of God by Mary (Lu 1:32-35), he says: “Jesus . . . being the son, as the opinion was, of Joseph, son of Heli.”—Lu 3:23. [italics theirs]
The full entry gives additional insights into Jesus' genealogical record.
As to the statement "Apparently in Judaic tradition back then, the legal father of a child was considered a father even if there is no blood relation.", I was unable to find a scriptural reference to a father adopting a son (or daughter) that would then receive tribal identity or inheritance.
The closest instance would be when, before the birth of Ishmael and Isaac, Abraham considered his servant Eliezer to become heir of Abraham's property.
2 Abram replied: “Sovereign Lord Jehovah, what will you give me, seeing that I continue childless and the one who will inherit my house is a man of Damascus, Eliezer?” 3 Abram added: “You have given me no offspring, and a member of my household is succeeding me as heir.” 4 But look! Jehovah’s word in reply to him was, “This man will not succeed you as heir, but your own son will succeed you as heir.” (Genesis 15:2-4 NWT)
In the topic of "Adoption" in the Insight on the Scriptures, it is noted that adoption was not a common practice due to levirate marriage:
Within the nation of Israel adoption does not appear to have been widely practiced. The law of levirate marriage doubtless eliminated to a great extent a basic reason for adoption of children: the continuance of the parental name.—De 25:5, 6.
[All scripture quotations from the New World Translation of the Holy Scriptures (Study Edition)]