Notice that Jesus chose 12 disciples (not 11 or 13). By doing this, by gathering 12 leaders reminiscent of the 12 tribes of Israel, many scholars believe that he was re-defining Israel around himself. It's as if Jesus said, "Israel, you didn't rise to the occasion of being a blessing to the whole world (Isaiah 2:2-4, as one example) so I'll take that job on myself".
Of course, the famous declaration of Peter, "Thou art the Christ (Messiah)..." prompted Jesus to say "upon this rock I will build my church". Now the interesting thing is that word "church". In the original language, the word is "ecclesia".
Now ecclesia is derived from "ek" - meaning "out of" and "caleo" meaning "called". So to translate ecclesia literally would be to say "called-out ones".
So the word "church" literally means "called-out ones". In the context of the New Testament, then, we would say that the called-out ones are a subset of both Jews and Gentiles that are "messianic", that is - they believe that Jesus is the promised Jewish messiah (that was to bless the whole world, by the way) and they follow Jesus as their master.
So no, we're not Jews in the sense that we follow the Jewish traditions - but we have been "grafted in" to one family of God, as Paul put it...