What does "Jewish" mean in that context? The dominant religion at the time was a sacrificial cult centered on Temple worship. rabbinical Judaism as we know it today developed over centuries after that temple cult was fully decimated by the Romans. Its completely anachronistic to say "Jewish" refers to Jesus in the same way that it refers to your local synagogue.
Right, Jesus is to be understood historically as a 1st century Jew who lead a movement. Just as Paul is understood historically as a very early Christian (and Jew), not a modern day one. Paul wouldn't have even understood something called Christianity as a separate religion.
