Jesus was no a "Jew" in the religious sense at all. If so he would have had no issues with the Pharisees. Jesus scolded their oral law and traditions which later got codified into the Talmud.
The Jewish religion from his day to today is not under Torah Law. And that is the argument - do the Jews, or anyone, have the ability to change God's Laws? The Jews did. And the Christians do too.
Jesus a Jew from the fact he was of Judea. Not by religion.