Yes it is a thing (albeit more subtle, which makes satire better or worse depending on your taste). I was able to read Chinese satire on U.S. gun shootings and PC issues quite frequently.
I guess my issue with things like this SP episode and these kind of satire in general, is that they only seem to focus on one important but partial aspect of one nation. To joke on American PC issues seems fun to many in China, but there are more important things to know about the U.S. than these things. In a similar vein I wish people interested in China do not just stop at knowing the terrible human rights situation and its increasingly authoritarian regime, but actually learn more about the shift of its regime since 1949, the more subtle parts of the ideology of the party (the ostensible resemblance to 1984 style "doublethink"), etc., which imo would provide much better insight into China than what is on HN.
I think GP was talking about satire of your own country, not propaganda about other countries.
Reminds me of the old joke:
Q: Is it true that there is freedom of speech in the Soviet Union, just like in the USA?
A: In principle, yes. In the USA, you can stand in front of the White House in Washington, DC, and yell, "Down with Reagan!", and you will not be punished. Equally, you can also stand in Red Square in Moscow and yell, "Down with Reagan!", and you will not be punished.
Well satire about Xi (and winnie the pooh) is also quite a thing, obviously censored in the most prominent channels, but still quite visible. I mean, I only say that because I assume the context ("i wonder if China has copied SP and if so how they would portray the USA") was SP satirizing China, so I offered the Chinese parallel.
And I just have to say it. The satire I am talking about comes from the actual citizens. They are not propaganda.
I think it would be too risky as crossing a line could mean prison.
All the Trump episodes on SP would definitely have ended with Trey and Matt locked up in China.