I think we need to remember that the remark of Pius XI(?) that spiritually we are all Semites was more than a rhetorical flourish, that a visit to a synagogue is more than a symbolic gesture. After all we worship a Jew.
And I agree...Christianity is really fulfilled Judaism, so we are also Jews, in a certain respect. And we should always be respectful of the particular people through whom God has chosen to reveal Himself, in the most special way imaginable.
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)