Yes he was a Jew, "King of the Jews" went through the Pharisaic Proselyte baptism because his mom already had him circumcized, offered the two turtledoves and when he was baptized according to the Pharisaic laws, he was officially a member of Judah.
Jew derived from Judah, the 4th son of Jacob/Israel.
that lineage Adam - Christ runs right through Judah (Genesis 49:10) because the scepter shall never depart from Judah
What Christ did was pour out his spirit over all flesh, Jew and Gentile both (Red, Yellow, Brown, Black and White Flesh)