It started in Florida with “Alligator Alcatraz.” Then came news of the “Speedway Slammer” in Indiana. Most recently the Trump administration announced plans for yet another immigration detention center, this one in Nebraska, to be called the “Cornhusker Clink.”

The alliteration, borrowed from the science of selling, goes down as easy as a Krispy Kreme for some Americans, for whom it is all in good fun. Memes have been made. T-shirts have been touted. The president has joked that detainees in Florida should learn the best way to run from an alligator, in the event of an escape from the center that opened last month in the Everglades.

“It’s got a ring to it,” Ron Buschelman, 66, said of the name “Cornhusker Clink” as he stood on Monday outside a farm supply store on the outskirts of Omaha. Referring to Mr. Trump, he added, “He’s got a very good sense of humor. And there’s nothing wrong with it. It’s a clink. It is. And we’re the Cornhusker State. I like it.’”

For other people, though, there was something repugnant about a government that would make light of an expanding mass deportation program that has sent immigrants to countries that they are not from, separated parents from their children, and deployed masked officers in unmarked cars to grab people off the streets.

“It’s not a reason to joke around,” said Roxana Cortes-Mills, legal director of the Center for Immigrant and Refugee Advancement, a nonprofit group in Omaha.

Still, ginning up indignation appears to be the point, at least in part, in this new era of government by troll. It is a strategy that the administration is leaning into in Mr. Trump’s second term — one that his administration is particularly fond of deploying in the realm of immigration enforcement.