getting one
A Cat License bought from the Zoologist and used once brings a cat in at dawn the next day. Nothing happens the moment you use it and there is no way to hurry it along.
what it does for a town
Pylons refuse to activate unless enough residents live nearby, and the cat counts toward that total while being ignored by the crowding math that makes real townsfolk unhappy about their neighbors. Dropping one into a cramped biome outpost is the cheapest way to make a pylon there legal without dragging a second person somewhere they will grumble about. It has no effect on anyone's prices.
housing it
It still needs a valid house and can be assigned to a particular one through its flag. It will share a room with a regular resident, so it does not cost you a building, but it will not share with another pet. Very small houses can reject it because it needs a spot of its own to stand in, and there may not be one going spare.
rerolling it
Breed and name are both rolled the moment it arrives, from six breeds, with no input from you. Using another license while one already lives with you swaps it instantly, puff of smoke included, for a fresh roll. Licenses are cheap enough that fishing for a specific look is a matter of patience rather than luck. Killing it also works, though it takes a few days to return and may come back as something else entirely.
the chair thing
Cats are the only town pet that will sit in a chair, which is worth knowing if you are decorating a room around one. Every other pet just stands. The pet option itself drops you into a fixed spot beside it and plays an animation that any input at all interrupts, and it needs a bit of flat floor in front of the cat to work.
the breeds
Black, Siamese, Russian Blue, Orange Tabby, Silver and White, each with its own pool of names, so the name it arrives with tells you which one you got before you even look at it. The bestiary artwork defaults to the Siamese.
the names
The list is a long run of references. Comic strip cats like Garfield and Hobbes, film and television cats like Jiji from Kiki's Delivery Service, Salem from Sabrina and Tigger, mythology in Sekhmet, Freya and Odin, games in Blaze from Sonic (given to the silver cat as a nod to her partner Silver), a cereal mascot in Tony, and at least one internet meme in Miette, which came from a post about a cat reacting with theatrical outrage to being nudged by a foot. Renne, Joshua and Estelle turn up across the cat, bunny and dog and all come out of the same role playing series.
the bestiary joke
The entry describes cats bringing presents that used to be breathing. There is no such mechanic anywhere in the game and nothing will ever be delivered to your door. The line is describing real house cats and nothing more.