Your comparison is hyperbolic for two reasons.
First, most universities are NOT censoring content or attaching warnings to texts. Period. This is obvious from the article.
Second, there's a difference between warnings and censorship. A few student groups are asking for warnings -- mostly about graphic accounts of rape in texts assigned in required courses -- not changes to the curriculum. Again, this is obvious and explicit in the article. And student groups can ask for whatever they want.
So, no. Even taking this article on face value, "this" (whatever this is) is not how "American universities "really are these days". And it's not even what student activists are asking for, by and large.
PS: Even if it were "this way", it's unclear to me what political movement such non-censorship is supporting. It's basically young rape victims asking to be warned when a lecture/reading is going to contain descriptions of rape. I'm not saying it's a good policy option, but excuse me if I roll my eyes when you compare it to a powerful tool of political oppression. Even the author of this opinion piece doesn't go that far. He basically just says "yeah there are jerks out there who are gonna make rape jokes, so you better grow a thick skin."