The hats’ appeal seems to rest partly in their studied outmodedness (think the 2.0 version of the trucker hats repopularized by millennials) and partly in their uncanny ability to capture the current absurdist political moment, with 17 Republicans vying, circuslike, for their party’s nomination.

“I’m at a loss to describe the ironic charm of the hats,” said Nu Wexler, a public policy spokesman at Twitter who received one as a gift from a colleague. “It’s a huge hat that looks like something you’d wear at a golf club in South Florida in the ’80s.”

Yet, Mr. Wexler added, “I’m a liberal Democrat who probably disagrees with Trump on every issue other than campaign finance, but I still respect his Twitter game.”

Louise Simpson, a junior at the University of Chicago who interned this summer at ABC News, gave the hats to her bosses as a parting gift. “The hats are kind of like a keepsake of the summer,” she said. “It’s a joke that everyone is in on, including Donald Trump.”