In order to get out of and possibly back into Paw-Tucket, pets need to have their own special key. Keys were first mentioned in Homesick as a Dog.
Mr. Yut explains that "personalized pet data" is loaded into every key. This is most likely basic information like the pet's name, their birthday, where they live, and so on. The pet also needs to provide a paw print (or whatever kind of print they have).
If a pet loses their key while in Paw-Tucket, they have to wait anywhere from a week to an entire month for a new one to be made. It's unknown whether or not keys are taken into the human world, or if there's any consequence to a key being lost there.