Yes, he was born one. I don’t think Jesus ever thought of himself as anything other than a Jew.
People came up with ‘Christian’ after he was gone.
There is the Jewish ethnicity and Jewish religions.
Yes, a plural. Before 70 AD there were multiple sects of Judaism: Sadduccees, Pharisees, Essenes, Jesus movement Jews and Samaritans. Sadduccees rejected the books of the prophets and the Oral Torah, the Pharisees had their Oral Torah.
Christianity is essentially a sect of 2nd temple Judaism.
After the temple was destroyed in 70 AD, onely the Samaritans and the Jesus movement Jews survived. The Sadduccees were touted to the temple so it was the end of the world for them.
It was also that way for the Pharisees but they created a new sect, created by Rabbi Yohannan bin Zakkai and that is rabbinical Judaism with the temple and priests and animal sacrifices replaced by synagogues and rabbis and todah.