On City Hall or an in-game computer, NRaas > SP > General Options > Options:Sims > Max Population > Cats > (try 1).
You can't set it for 0 here as mentioned earlier because to the mod 0 means no cap. You could try -1, not sure if the mod will read that correctly, but 1 should be safe if you don't mind one resident cat somewhere in the world.
Overwatch has an option to shut down the formal Neighborhood Pet Adoption Centers, this is a random inactive household that picks up the designation that will suddenly have baby animals to give away. The mere existence of SP shuts it down also. The Adoption Center household scenario often gets stuck and breaks, and the baby animals magically appear with no genetic connection to other animals in town so players tend to not like it very much. It has nothing to do with regular pet adoption from the virtual shelter.