The point about being Jewish - in common with being Muslim, is that it is no just an aspect of life, as it is in Christianity, it's part of the fabric of society and has a far more direct influence and effect on its people than the Christian faith.
Judaiism influences language, food, and culture far more than Christianity - which is why they are more prone to mention their Jewishness. It's part of what and who they are, so it's far more likely to produce references, oblique and direct, in conversation.
You can make fun of everybody and everything - the only consequence depends on your audience at the time.