The first Christians were Jews. Modern Arabic speaking Christians (With the exception of the Copts) and Neo-Aramaic speaking Christians are descendants of the first Jewish Christians.
Modern Judaism also isn't the same as 1st century Judaism. Several times they have changed in reaction to Christianity, such as rejecting books of the bible that they formerly considered holy. At one time they used the Septuagint but after Christ came they rejected it, and a number of books (the "Deuterocanon") written in Greek solely because they were written in Greek during a time that most Jews spoke Greek, despite God never stipulating that his prophets would only speak and write in Hebrew or Aramaic.
Christianity is a "Jewish religion" even if the vast majority of it's members now are not Jews. It didn't come from Rome, Greece, England, Egypt or anywhere else, it came from Jews who apparently did not find it impossible to believe Christ represented the culmination of their religion, which they were better authorities on then anyone today, including modern Jews, could ever be.