Cats are mysterious creatures, so it was inevitable that someone at sometime would create stories featuring cats involved in crazy, quirky, entertaining mysteries. Presumably the first to do so was Lillian Braun Jackson, who came out with The Cat Who Mysteries in the 1960s, before the term cozy mysteries was coined.
Now there are dozens of cozy mystery series with cats 
along with cozy mysteries featuring quilting, cooking, baking, libraries, travel, dogs, and others. While this genre (or sub-genre) has guidelines—some of them have gone out the window as authors create their own themes and styles and what they 
consider appropriate for cozy mystery readers.
Some authors feature cats that talk and have a point of view, some main characters are detectives, amateur sleuths, veterinarians, librarians, bakers. Yes, we’ve stepped outside the box and that seems to 
be okay with readers, as long as the cat (dog, horse) has plenty to do in these stories and the stories are fun and entertaining and provide little surprises along the way.
I know that many of you enjoy my Klepto Cat and Calico Cat Mysteries. Here are a couple of sites listing others you might want to read. There’s no such thing as too much fun cat reading is there?
https://cozy-mystery.com/blog/kittens-cats-and-cozies-cat-mysteries/
https://cozy-mystery.com/blog/some-of-the-best-cozy-mystery-series-that-feature-cats/
OMG, what a lot of new cat cozy mystery series I just found. Between the two articles there are quite a few I had not known about or not yet read. Braun’s books were probably my first exposure to cat cozies, although there was no such term then. I have many of her print books stored away, but will probably start over with her e-books since it has been so long. I was surprised that I had already found and read many of the series mentioned, but there always seems to be more to start! Oh well, more for my TBR pile again.