At minimum, perhaps he’s fantasizing about joining the ranks of autocratic leaders like Turkey’s Recep Tayyip Erdogan, whose government prosecuted people for sharing memes of him as Gollum, and China’s Xi Jinping, who cracked down on comparisons of him to Winnie-the-Pooh.

It may be that this is all just blowing off steam, the virtual equivalent of Elvis shooting his TV. Maybe it’s a combination of authoritarian role-play and wounded celebrity ego. Maybe the federal authorities can’t or won’t act on the president’s yearning for “retribution.”

But that doesn’t mean no one else is listening. Last fall, explosives were mailed to prominent political critics of Mr. Trump, the offices of CNN and the actor Robert De Niro, who assailed Mr. Trump in a speech at the Tony Awards. Mr. Baldwin said that Mr. Trump’s Sunday tweet — echoing his labeling of the press as the “enemy of the people” — could be “a threat to my safety.”

Mr. Trump’s complaint is also an oddly sensitive reaction for a man who not only ran for president as a tough guy but spent decades being satirized as a media figure.

Then again, that may be exactly why it stings him so deeply now.

Mr. Trump is used to dealing with “S.N.L.” as a celebrity, someone who might be mocked but would be in on the joke. He’s been a recurring character on the show going back to the 1980s, when Phil Hartman played him in a holiday sketch that embellished his brand by rendering his name in gold lettering.