Choice of the Cat.
The PC of New Witch in Town has an animal familiar (dog, cat, or bird), and they can befriend various NPCs who also have pets.
In Royal Affairs, the PC has a dog, a horse, or a bird of prey as a pet.
In Social Services of the Doomed, you can have a cat or a fish (or a pet rock).
You have a cat in Out for Blood. You can have an animal companion in Night Road (wolfdog) and Parliament of Knives (rat) if you choose the right clan background. You can befriend an animal (dog, cat, bird, or deer) in Sins of the Sires.
Runt of the Litter and The Last Monster Master involve training fantasy animals.
Zombie Exodus: Safe Haven allows you to have a dog or cat.
Atlantis Academy and Lies Under Ice, both coming in November, include animal companions. In Lies Under Ice you can choose to take cats or rats as companions when establishing a space colony. A companion water dragon is a major character in Atlantis Academy.
Fine Felines by Felicity Banks is about breeding cats. It was written for IFComp 2021 and will eventually be published by HG.
There are NPCs in The Play’s the Thing and Stars Arisen who have dogs, and there are even Achievements for interacting with them.
You didn’t say anything about Heart’s Choice, but a dog is a significant character in Brimstone Manor, you can adopt a kitten in Belle-de-Nuit, and there’s an Achievement for conjuring an ethereal puppy in Forbidden Magic.