It's almost like, on some level, they don't actually believe that they're the downtrodden ones sticking it to those with power. Like this is just a talking point that's used to justify the actions they're trying to take with the power they claim not to have, and to dismiss any concerns about the morality and consequences of those actions as concern trolling - after all, if you really cared you'd be worrying about what the people with power were doing, not trying to police the powerless - and then thrown aside when it's not useful.

What makes this particular approach really dangerous, of course, is that the easiest - and perhaps only - way to enact this censorship is if the people demanding it actually do have power within the existing system and aren't particularly oppressed. There's some truth to the oft-repeated argument that this can't be oppressive because of power structures - if, say, a bunch of woman of colour who're prominent for their activism within their community think this way, that affects no-one. If a chunk of the existing white male political establishment latches onto this for their own goals (like, say, defeating the Republicans), on the other hand...

Both the left and right (and centrist!) authoritarians are only annoyed when things that they agree with get censored. This is the whole point of authoritarianism after all, that you do not tolerate other's opinions.