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Max Schachter likes to dress up as the very person who seemed to taunt him — wearing a red jacket emblazoned with an eagle and a “Make America Great Again” hat — at a demonstration in New Jersey on the evening of Oct. 26. On that night, Mr. Schachter witnessed a clash of protests over the detention of immigrant children by immigration officials, a small group of Republicans yelling across the street, yelling at the other side to “leave our children alone.”

There was Mr. Schachter, dressed like a protester and passing out fliers about an event at a Connecticut conference featuring conservative columnist Ann Coulter. When an off-duty police officer flashed his patrol car’s lights at them, it stopped their march.

“It was ugly,” Mr. Schachter, 25, said. “I have some protective glasses in my car. It’s stupid, but I’m going to dress like that. I’m sticking up for my conservative brothers and sisters.”

To this day, Mr. Schachter wears his mask, a faceless tie-dye version of the red jacket and a red beard at everything from his family events to holidays at his new job as an internet communications officer for Hillary Clinton’s 2016 presidential campaign. Mr. Schachter, a white New Yorker, in no way subscribes to Mr. Coulter’s conservative views, though he generally opposes the political correctness he says is a plague on public life.

For years, his regular go-to costumes were his mother’s skeleton dress and his father’s green-and-red bow tie. Mr. Schachter’s friends would sometimes mock him, sometimes threatened him, and sometimes he said he believed he had been attacked at the meeting last fall.

None of it made sense to him. But he thought people felt threatened enough that they attacked him.