I tell my sim to search adoptable pets in the neighborhood, but it always says horses or puppies...

Not sure which mods you might or might not be using, but if NRaas StoryProgression is among them (no options here, just its mere presence does this) or Overwatch with the too often broken Neighborhood Adoption Center scenario switched off, then the only way to get baby animals into a world is for adult animals to do what comes naturally to them and breed or find them though the pet shelter ads, but not from other households unless those sims are allowed to adopt pets themselves. Are there any adult cats around town? If not, then the places to find cats would be the pet shelter rosters available through the phone/computer/newspaper, and the strays which are never supposed to be babies. Can you maybe place some adult cats in other households and hope they find each other attractive?

If you aren't using those mods and are relying on the spontaneous creation of baby animals by way of the Neighborhood Adoption Centers spawning, then afraid I wouldn't know why cats are being excluded from that one.

Well, none of my households has ANY pets yet, so I'm not sure why even baby dogs and horses keep popping up.
But then maybe I will add cats to a couple of houses.

Maybe it would help clarify a little if I said this the other way around? If you do not have NRaas SP, nor Overwatch with the EA Neighborhood Pet Adoption Center switched off, then you have that Center in play. This is in addition to being able to adopt pets from the shelter by way of the phone, etc.

Periodically, the way this is supposed to work, an inactive home is targeted and announced as the Center with some baby animals to give away. The residential lot gets a special marker on it in Map View. These baby animals just magically appear, they have no parents so no genetic ties to any adult pets in town. If no one, active or inactive, adopts the baby animals, I believe they just disappear after a while and we are supposed to assume they were adopted to out of towners. Many players do not like the randomness of this method and the scenario unfortunately gets stuck often enough with some strange side-effects. I've never heard of the Centers forgetting to include kittens, but if you were relying on these in the past it's not that much of a stretch to think that could now be not working as intended either.

Wait. Actually I have the Overwatch mod. I did not alter any of the settings, so it is on default. With this in mind, what is up with the adoption? Sorry for my misunderstanding and all. >_<