Arjen Lubach, a Dutch late-night host whose show, “Zondag met Lubach” (“Sunday With Lubach”), made the video that started the trend, said, “Because we realize it’s better for us to get along, we decided to introduce our tiny country to him in a way that will probably appeal to him the most.”

The English-language voice-over on this video belongs to a Dutch-based American comedian named Greg Shapiro. “I’ve been working on the imitation for a while,” Mr. Shapiro said in a phone interview on Sunday from Paris. All it takes, he said, is repeating Mr. Trump’s interviews or tweets verbatim. “There’s hardly any heightening necessary,” he said.

After the video went viral, Mr. Shapiro said, “it was nice having family from America send me these clips saying, ‘Oh, this is like Dutch and Trump.’ And I’d say, ‘Yeah, that’s my voice.’ My mom did recognize it.”

Mr. Shapiro, 48, has seen a bump on his YouTube channel, where he offers lessons in Trump impressions on his “United States of Europe” show and posted a spoof news conference on Monday,in which he impersonates the president.