getting one
A Dog License from the Zoologist, used once, brings a dog in at dawn the following day. Nothing arrives at the moment you use it, and the license is consumed either way.
what it does for a town
It counts toward the number of residents a pylon needs while sitting entirely outside the crowding calculation that decides how happy everyone is. That is the whole mechanical reason to own one: it props up a remote pylon without making a neighbor miserable or pushing anyone's prices around. Everything else about it is decoration.
housing it
It needs a valid house like any resident and can be pinned to one through its flag. It will share a room with a regular townsfolk, so it costs a bed rather than a building, though two pets will not live together. A very small room can still turn it away, because it needs its own patch of floor to stand on.
rerolling the breed
Six breeds, rolled at random when it moves in, one at a time in the world. Using a second license on a dog you already have exchanges it immediately for a fresh roll, smoke puff and all, which is the only way to steer the outcome. Killing it also resets things, but it takes several days to come back and it may return as something different.
petting it
Talking to it offers a pet option that positions you beside it and plays a short animation. It does nothing beyond looking nice, needs flat ground in front of the dog, and cancels the instant you use an item or take a hit.
the breeds
Corgi, Labrador, Beagle, Dalmatian, Pit Bull and Husky, each with its own name pool, so the name tells you which one turned up. The bestiary picture defaults to the Labrador.
half the names are memorials
A lot of the list is personal rather than referential. Sparky was the lead developer's own dog and was originally going to belong to the husky instead of the labrador. Pebbles was added later for a player's labrador. Navi and Zelda belong to the developers as well, doubling as a nod to the games they came from. Shino and Kona both belong to a beta tester, and Kona is the one he had to leave with his parents when he moved abroad.
the rest of the names
The others reach outward. Balto was the real sled dog on the serum run to Nome, and Hachiko the Akita who kept turning up at the station long after his owner stopped arriving. Indiana comes from the childhood pet that gave Indiana Jones his name, which is why it sits on the husky. Marshall covers both the PAW Patrol character and the long tradition of dalmatians in American fire houses. Odie is out of Garfield, Spike out of Tom and Jerry, and Sven is what a very well known streamer called his wolf in another game.