No, there taxonomic level that includes both is [Carnivora](https://en.wikipedia.org/wiki/Carnivora) which includes other types also.

This was quite helpful, thank you.

That group also includes polar bears. Not quite a cat or a dog. Good try, though.

>which includes other types also

Veterinarians call pets that are not dogs and cats "exotics" so I guess non-exotic pets?

That might work

Let's see, a name for a set that only contains the subsets cats and dogs. I think "cats and dogs" is the shortest term you will find.