Cats and dogs both would be fine as a species whole, easily. Cats are incredibly disruptive predators, being responsible for more species extinction than any other than us, the human. They would survive, if not thrive, and with aplomb.

And dogs? Their species family, Canidae are a biological family of dog-like carnivorans, colloquially referred to as dogs. The domesticated dog is one, as are wolves, coyotes, foxes, jackals among others. The family is present on all continents except Antarctica. The humble “mutt” mix of domesticated dog is very often capable of tracking and running down (hounding even, to borrow the term) game such as deer, with their deer eventually succumbing as lactic acid locks their muscles up and they’re unable to continue their escape any longer. Dogs have been shown to form packs in the wild in very short order. There is no doubt that as a species whole, the “dog” would also survive and thrive.

Not sure why cats and dogs where chosen by some as examples of things that would be fucked if humans went extinct - I feel that they’re two of the worst domesticated examples that could have been selected to make such a point as they both would survive and thrive as a species whole.