There's plenty of dogs to be found around Skyrim, but where are the cats? That's something that several mods have sought to fix over the years, yet the latest introduces some more complicated adoption and needs systems to make sure you're furry immersed.
Having been in development since 2018, A Cat's Life is a mod by Darkfox127 releasing on 1st August for both Legendary and Special Edition Skyrim. The mod allows players to adopt cats from most major cities, but things aren't quite as straightforward as just asking: some of the stray cats need to be convinced. You can observe cats to see whether they are friendly, then improve relations by offering food or going in for a pet. Be warned, however, that the latter is a risky strategy that can result in some scratches.
An easier route is to simply buy a feline from one of the two rival pet stores added by the mod (complete with new characters and voice acting). You can technically buy all the cats in the stores, but you can only have five cats follow you in total - otherwise you would probably become some sort of feline pied piper. The mod does provide a perk called "crazy cat person", although exactly what this does - or how you achieve it - is a bit of a mystery.