Jesus was not a Jew in the "religious sense". He was a Jew for the fact he was of the tribe of Judah. That is 1 definition of Jew as there are many.

If Jesus was a "good Jew" he would have bought into what the Rabbis have been teaching going back to Solomon. Jesus called it the "customs and traditions of elders that make the Law of no effect". Today it is called the "oral law" or Talmud - which codified this oral law into the basis of modern day Judaism NOT base don the written law of Moses or Torah.

The Jews are not under this written law. and when Paul left them he left the oral law and went back to the written law. As Paul says…

Act 24:14  But this I confess unto thee, that after the way which they call heresy, so worship I the God of my fathers, believing all things which are written in the law and in the prophets: 

Jesus and Paul are what you would call a "Spiritual Israelite" not a Jew.