The word Jew implies someone who follows the modern religion of Rabbinic Judaism.
Jesus Christ condemned the Rabbinic Judaism of His times for its hypocrisy and voiding of the Law. He certainly did not follow them.
Moreover, Blessed Mary was a Galilean, not a Judean (St. Joseph was a Judean from Bethlehem, but St. Mary was from Nazareth). The Jews had not lived in Galilee since the time of the Maccabean revolt (1 Maccabees 5.14-23). The people there who followed the Law were not genetically Judean, but were the mixed remnant of the 10 tribes of Israel and the gentiles who had moved in. Thus the utter contempt of the Rabbinic clique of Jesus' day for Him because of His Galilean roots (St. John 7.52), and even some of his own Apostles before they met Him (St. John 1.46). After all, it was "Galilee of the Gentiles" (Isaiah 9.1, St. Matthew 4.15). Galilee was a microcosm of the world, with all manner of ethnicities living amongst each other - Greeks, Arameans, Phoenicians, Israelites.
Jesus was purposefully born in Galilee of a woman of unknown ancestry because He came for all people. He certainly was not some Yeshiva Rebbe hunched over a desk muttering Talmudic verses.
If He is to be said to be anything, He should be called a wandering Aramean, like His father Abraham (Deuteronomy 26.5, St. Luke 1.55)
You have a different take on history than I do. In my view, rabbinical Judaism was established after the destruction of the temple, a development of certain strands of the Pharisees.
A woman of unknown ancestery? Galilee was a frontier region for the Hasmonean government. It may be speculated that Joseph was a native of of means who had relatives in Nazareth, of whom Mary may have been one. In any case, Mary was a woman of Israel, otherwise Jesus was not a Jew, which is, of course what you choose to believe. It is strange therefore that he limited his ministry to the Jews, with the exception of his mission into Samaria. He did say that salvation is from the Jews, unless you are trying to spin that as an ironical statement.