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.
The Talmud was written over several centuries after the fall of Jerusalem. They had failed to keep the law because God didn't intend it to be be kept outside of Jesus. So, they sat down and tried to interpret it themselves without Jesus. That was impossible. Their interpretation got pretty far fetched, in some cases to the point of nonsense.
Sure, it is that, but it's also the later codification of the Pharisaical/Rabbinical oral traditions condemned by Jesus, which he referred to as the doctrines of men. What we know today as Judaism is essentially the offspring of precisely what Christ condemned in the Pharisees. The Pharisees basically took over charge of those Jews who rejected Christ, and manufactured a new religion after the fall of the Temple.
