>Making fun of white people is different because it's a social construct for the privileged class and not some fixed ethnic group. It's a critique of power and not a group of people.

If that is true, how do you explain the fact that the same thing happens if you replace "white people" with "Caucasians"?

Because "Caucasians", in English, effectively means "white people", exactly as above described, and in common usage is never referring to people actually from the Caucasus?