Before the Europeans came to this continent to bring forth a new nation, the native people did not have cats. There were several different kinds of wild cats, most of which survive today, but none of them were domesticated on this continent. The Europeans kept cats as companions and/or vermin hunters. Often ships would have a cat or cats on board to kill the rats. It is possible the first cats came to this country as sea cats, jumped ship, and found themselves in a whole new world.
Cats have probably been domesticated for around 10,000 years, and are descended from a small desert wild cat. The first evidence of human interaction with cats (common burial) is from Cyprus. We thought we were domesticating the cat, the cat thought it was domesticating us. It turned out to be a partnership. Cats do not have a pack instinct like dogs do, and therefore do not accept humans as the pack leader, or alpha cat. If you have ever been to MarshallO’s profile page, his bulletin board says it all: "Dogs have owners, cats have staff."
Based on DNA evidence, the wolf ancestors of modern dogs diverged from other wolves about 100,000 years ago, and dogs were domesticated from those wolf ancestors about 15,000 years ago. This date would make dogs the first species to be domesticated by humans.
Evidence suggests that dogs were first domesticated in East Asia, possibly China, and some of the peoples who entered North America took dogs with them from Asia. (The preceding paragraph is from Wikipedia.)
So there is folklore among American Indians about dogs, because they brought them here with them over the Bering Land Bridge, and by that time the humans and dogs had a long history together. It is quite likely that they used dogs as a type of pack animal as well as a hunting companion and protector.
Now close your eyes and try to imagine some way you would get a cat, or group of cats over the Bering Land Bridge. Not a pretty picture is it? They would have to be carried, and the people coming over here had more important things to carry. And the pack dogs would probably have objected. The cats wouldn’t have liked it either. Besides, the continent was pretty well closed to all but sea migration by the time the cat was domesticated.
So the native people of the Americas didn’t have a lot to say about cats because they weren’t that familiar with them.
I’m sorry, but I don’t have any recommendations for where to research this further. I have studied preliterate societies and I can’t remember that there is any folklore about domesticated cats in any of them. Cat Fancy seems to be a byproduct of civilization and writing. Of course, there are a lot of things I can’t remember.