FWIW, I first encountered the phrase as the punch line to a different joke.
A Jewish guy goes to Tokyo on business. He is observant, so on the Sabbath he finds the only synagogue in Tokyo. He particpates in the service, and then is chatting with other members of the congregation afterwards. He then wishes the rabbi good-bye and leaves.
And the (Japanese) president of the congregation says to the (Japanese) rabbi, “Nice guy. I hope he comes again.”
“Yeah”, says the rabbi, “but it’s funny - he doesn’t look Jewish.”
I was once at a presentation when a black guy introduced a long time family friend and Jewish lawyer as “Uncle Shelly.” When the person next to me said “Uncle Shelly?” I replied “Yeah. He doesn’t look Jewish.”
The person I was talking so started laughing so hard he had to leave.
The Chipmunks used this with “Irish” instead of “Jewish” in their rendition of “My Wild Irish Rose”. Between sections of the song, Alvin asks Dave if he’s Irish. The final exchange interrupts the song just before the last line.
Well, you don’t exactly have to drop your pants at the door (what would the feminine equivalent be? submit receipts proving you’ve never paid full retail?)…but yeah, Broadway is the one place you know a good % of your audience will be of the persuasion.