People with pets dispose of their deceased animals’ remains in different ways, but if they live in the city the options are obviously more limited. You can’t just go in the woods and bury Pochi or Tama and set up a little memorial. Moreover, there are laws about disposing of dead animals.
About a year ago our cat died. We’d lost two cats previously. With the first one, we called a local temple, which immediately sent someone over to take the body away. Several days later they called and we went to the temple where they said a little prayer and gave us the cremated remains in an urn, which we took home, even though the temple has a reliquary for pet remains. For the second cat we called a pet cremation service directly.
Most such services are associated with temples so as to make their work seem less commercial, but even when you take your pet’s body to a temple they send it out to a commercial cremation facility anyway. These services will pick up the body at your home and later bring back the ashes; or, more precisely, the bones, since cremation in Japan—even for humans—doesn’t usually get as far as ashes, which entails another, different cremation process.