The term Jew is used in at least two senses in Scripture: to refer to those who are ethnically Jews and to those who are religiously Jews. Jesus was a Jew in both senses. In fact, he completed the Jewish religion by serving as the Messiah (Christ) whom the prophets had long foretold.
The completed form of the Jewish religion is known as Christianity, and its adherents are Christians or "followers of the Christ." Unfortunately, many people who were ethnically Jewish did not recognize Jesus’ role as Messiah and so did not accept Christianity, the completed form of Judaism. Instead, they stayed with a partial, incomplete form of Judaism. Other Jews (the apostles and their followers) did recognize that Jesus was the Messiah and embraced the new, completed form of Judaism.
Shortly thereafter it was recognized that one could be a follower of Christ even if one did not ethnically join the Jewish people. Thus the apostles began to make many Gentile converts to the Christian faith. It is thus possible for a person to be a Jew religiously (because he has accepted Christianity, the completed form of the Jewish faith) but not be a Jew ethnically. This is the case with most Christians today.
It is this difference between being a Jew ethnically and religiously that lies behind Paul’s statement in Romans 2:28-29: "For he is not a real Jew who is one outwardly, nor is true circumcision something external and physical. He is a Jew who is one inwardly, and real circumcision is a matter of the heart, spiritual and not literal."
Christians are those who Paul refers to as being inwardly (religiously) Jewish, while non-Christian Jews are those who he refers to as being outwardly (ethnically) Jewish. The former condition, he stresses, is the more important.
Unfortunately, over the course of time some Christians broke away from the Church that Jesus founded, and so a name was needed to distinguish this Church from the ones that broke off from it. Because all the breakaways were particular, local groups, it was decided to call the Church Jesus founded the "universal" (Greek, kataholos = "according to the whole") Church, and thus the name Catholic was applied to it.
That is why Jesus was a Jew and we are Catholics: Jesus came to complete the Jewish religion by creating a Church that would serve as its fulfillment and be open to people of all races, not just ethnic Jews. As Catholics, we are those who have accepted the fulfillment of the Jewish faith by joining the Church that Jesus founded.