He was adopted by Joseph and raised as his son. Under Bibllical law, that makes him a Jew.
And his biological mother was, of course, a Jew, in the sense that the term meant at that time.
Today, Jew means someone 1) claiming a special privilege before God stemming from physical descent from Abraham, 2) who rejects Jesus, and 3) who considers the Talmud to be a collection of sacred texts. In other words, Jew today means something quite different than it commonly meant when Mary was born and raised as a Jew, and is more correctly termed Rabbinical and/or Talmudic Jew, which essentially amounts to what Christ would have referred to as a Pharisaical Jew, implying a profound and demonic unfaithfulness to divinely delivered Sacred Scripture, the Patriarchs, and the Prophets.
