Jesus was not a Jew in the modern sense of the term. He was a son of David. The term Jew does not even appear in Scripture until hundreds of years after David, to prevent this character assassination. He was not even a religious Jew, but an Essene, and had light colored hair and eyes, according to the letter of Pilate to Caesar.
The term Jew is even distinct from the Judahites (sons of Judah) referring instead to the citizens of Judea after the Pharisaic Edomites drove the earlier priesthood from the city.