Reference to a dictionary will reveal that "satire" is, broadly speaking, a work using humor in such forms as irony and exaggeration to ridicule the vices and stupidity of individuals or entities. So defined, satire has been around for thousands of years. In the West, when ancient satirists are considered mention is normally made of Aristophanes among the Greeks, Juvenal, Petronius and Lucian among the Romans. Martial as well, perhaps, though his satire was far more condensed being in the form of epigrams. Seneca too, among the Romans, at least for his ridicule of the Emperor Claudius, applying for membership in the fellowship of the gods after his death. There have been many other satirists throughout history--Rabelais, Voltaire, Cervantes, Swift and Poe, for example, are some of the more famous of the lot.
Satire is generally thought of as a kind of literature, and those named just above are of course writers. Sometimes, though, drawings and paintings are satirical, like the print by Gillray at the top of this post, called The Plumb Pudding in Danger. The plumb pudding is, obviously, the world and it's being carved up by William Pitt, Prime Minister of England, and Napoleon, who had by then become Emperor of France.
The quality of satires varies considerably. So does the kind of humor or style of writing employed by the satirist. Aristophanes' humor was somewhat farcical, as in his lampooning of Socrates and his circle The Clouds. Lucian also targeted philosophers and philosophy, as well as other people and things, and was a mocker. Juvenal's humor was dark and sharp; he had no light touch (I think of his satires as more in the way of rants). Seneca's satire on Claudius struck me as more cruel than witty. Rabelais seems to have made use of absurdity in Gargantua and Pantagruel. Petronius' humor was deadpan, in my opinion. His Trimalchio is unquestionably vulgar and pompous, but Petronius makes this evident merely by describing his statements and conduct, rather than commenting on them. Irony seems to have been Swift's weapon of choice.
In these dark times, there's a concern by some that wit and humor, which would include satire, are being restricted if not entirely eliminated by the tendency of some others to condemn those who use them in referring to certain people or groups of people who are perceived as being or having been treated unjustly because of their difference from what's considered normal. The most recent example that comes to my mind relates to the character Apu in The Simpsons and the apology made by Hank Azaria (his "voice") for his contribution in the stereotypical (though intended to be humorous) depiction of the character running a convenience store. The apology caused John Cleese, formerly of Monty Python, to apologize for mocking white people as part of that group.
Cleese's point is fairly obvious, I think--that humor, and satire, are used in reference to all people indiscriminately, and should be. In other words, that no one should be exempt from mockery merely by virtue of what they are.
That makes a certain sense, as does a related claim being made in these dark times--that nobody should be discriminate against, and therefore treating certain people better than others because or their race or religion, for example, is wrong. From that, of course, it follows (or so it's claimed) that white, Christian people are being discriminated against when others are given certain advantages or benefits, or are deemed to have special claims to be satisfied.
In order for that position to accepted as appropriate, however, it's necessary to assume that we all are in the same position when it comes to treatment by each other, society and government. If we existed on the fabled "level playing field" then a case can be made that favoring one group or person over another merely because they have a certain skin color or religion or other characteristics would be unjust. That playing field doesn't exist, however, and never has existed. To pretend that it does for purposes of complaining about discrimination against currently privileged people is fatuous, and indeed unworthy.
When we insist that we should treating each other as if we all existed on a level playing field, we acknowledge that it would be desirable that we should live on that level playing field. If we acknowledge that, though, we have to admit that we don't at this time, at least if we're honest. So, one would think it would also be desirable that levelling steps should be taken, But that would mean that those underprivileged or discriminated against unjustly, here and now, be benefited more than those who are not. Unfortunately, that's not something the haves are normally willing to grant the have-nots, particularly if that would mean the haves would have less than they do now.
So, I don't think this is a viable position, even when it comes to humor. Does that mean that humor, and satire, will inevitably disappear?
I doubt it, unless we expect that the kind of totalitarian society envisioned by the notoriously humorless Plato and put into place by such as Stalin and Mao will be our future. The great Roman satirists lived in Imperial Rome, not exactly a free society, and managed to thrive. I don't think shaming of the kind being practiced now will significantly restrict our tendency to mock and ridicule each other. As well expect that wars will end. Perhaps the more egregious kind of ridicule will be reduced, which wouldn't be a bad thing. Satire is best when it is witty and subtle, not blunt and obvious or offensive. It will survive all the moralists of this time as it has in the past.