Not so. Jesus was a Jew. The Jews who believed that he was the promised Messiah remained Jews. When the Gentiles or pagans and all non-Jews came to believe that He was the promised Messiah and Lord, they became “followers of Christ.” Eventually, when the Temple was destroyed in AD 70, the religious “Jews” who did not accept Jesus essentially redefined the meaning of what it meant to be Jewish. In a nutshell, it was “We are NOT people who believe Jesus is the Messiah.”
The whole concept of Christian is a follower of Christ. Christ does not follow himself, so he is not a Christian.
Christians are those who follow the Annointed One and Jesus IS the Annointed One.
Well, the word “Jew” refers both to religion and to race. You can be a secular Jew today, for instance, although I would argue that that is to be an incomplete Jew. Jesus as a human being was Jewish, since his humanity was born of Mary, who was Jewish.
As you suggest, Jesus was also the promised Messiah of the Jews, the fulfillment of all the Prophecies. He fulfilled the promises of the Old Testament. And He established a New Covenant. But in the O.T., God tells the Jews that His covenant is with them “forever.” And in one of the Epistles, Paul says the same, echoing those words.
Christianity did not replace Judaism, it fulfilled it. Those who remain Jewish refuse to recognize that. But God remains with them as long as they are faithful according to their lights. So He has promised.
So, is Jesus a Christian? Well, traditionally, the Church is the Bride of Christ, so you wouldn’t exactly say that Christ is a member of that Church. He also is the Vine and we are the branches. So, kind of hard to say, but I would say, yes. As Christians, He is our primary model, after all.