I'm not sure whether that's true of every minority identity. As a Jew, for example, I feel connected to other Jews by the great lingua franca of sarcasm. Oh, I'm sure there are Jews who aren't sarcastic -- I just haven't met them.

My former brother-in-law is a gay Republican. I'm not sure whether that affiliation has anything to do with protecting "privilege" -- a word I suspect he wouldn't understand in the sense you're using it here. I think he just doesn't like change.