TOKYO—On their recent honeymoon in Japan, Jason and Nicole Francis savored the temples of the ancient capital of Kyoto and unwound at a resort with a private hot spring. Later they dressed up as a lime-green dinosaur and an Italian plumber and sped through the streets of Tokyo in go-karts.

“I picked Yoshi because I always pick Yoshi in the game,” said Mr. Francis, a 35-year-old emergency-services worker in New Jersey, explaining his decision to don the dinosaur costume.

The...