I've read that the Aztecs kept dog as pets (the ancestors to the modern chihuahua breed), but I haven't heard about anything else. Domestic cats would have been unknown before European contact, but are the Aztecs known to have kept any other animals as pets?
This previous topic may answer your inquiry,
But if you're talking about non-elites, the only "pets" commoners may have kept were turkeys. But like dogs, turkeys were often raised as sources of food either in the form of eggs or the bird itself. Turkeys were also kept for their feathers, which could be quite colorful.