KAWASAKI, Japan—Yahui and Tatsuo Matsui met because of their dogs, Ai and Doggy. So, at their wedding reception, the couple dressed up their pets as a traditional Japanese bride and groom and seated them at the head of the table.

That was in 2004, when Ai and Doggy were still pups. Now they are 15 and 13 years old. Ai’s stride has slowed and Doggy’s neck is so stiff he can barely move his head. “Whenever Doggy finds Ai, he tries to greet her but she rarely responds,” Mr. Matsui said....