SEOUL, South Korea — In a country that has faced political divisions and difficult relations with its nearest neighbors, the popularity of an item related to the president has made headlines.

No, not President Trump’s $25 “Make America Great Again” cap.

It’s the President Moon Jae-in watch, an unpretentious timepiece with a gold-colored case and beige sheepskin strap, named for the South Korean leader. The value is about $40, but it’s not for sale — the quartz timepiece is given to guests at the presidential Blue House, to foreign dignitaries and to Koreans living overseas on occasions when the president visits.

The tradition began with President Park Chung-hee, who was in office from 1961 to 1979. The first watch was made during Mr. Park’s administration, but the presidential Blue House did not provide the exact date. Some collectors said it was for participants in the Saemaul, a rural-development political initiative, in the 1970s.

Regardless, watch collectors and political supporters alike have become fans of the keepsakes.

“I heard they’re very inexpensive and produced for a very small amount of money, but they actually feel quite luxurious,” said Hong Jeong-hwa, 29, an attorney and a member of Incheon Metropolitan Council. “They don’t look cheap.”