Yes, there is a common ancestor to the cat and dog. If you believe in evolution as set out by Charles Darwin in his work “On the Origin of Species by Means of Natural Selection…” you will believe that there is a common ancestor to all animals not just cats and dogs. And that first common ancestor experts have decided is a sea sponge which existed about 640 million years ago and still exists today.
Around 57 million years ago experts have also suggested that the common ancestor to cats and dogs existed. They’ve named the animal Dormaalocyon latouri. They say that it looks like a cross between a tiny panther and a squirrel. It was a tree dweller weighing about 2 pounds.
Evolution led to many species evolving from this very early animal of which were the group of animals that we now call Carnivora (flesh-eating mammals). And this group of animals categorised as Carnivores include dogs and cats. The chart below shows this evolutionary route. They decided that the Carnivora first evolved at around 41 million years ago.
They found the remains of this common ancestor in Belgium. There is no doubt that this theory is speculative. There is no certainty it seems to me that this is indeed the single common ancestor of cats and dogs. But what is absolutely certain is that there is one animal and it lived a very long time ago and through evolution we have this multitude of species called carnivores and in that very large group of animals there are cats and dogs including domestic cats and dogs.
They say that modern carnivores evolved from a single group known as carnivoraforms which existed in the Paleocene (66-56 million years ago) and Eocene (56-33.9 million years ago) periods.
The chart below shows the divergence of Carnivora:
The Carnivora evolved from one of the miacids which split into the Caniforms (dog-like creatures) and the Feliforms (cat-like creatures).
Note: the categorisation of the species is called taxonomy and it is constantly evolving. It is still not a settled area of science.
Sources: Daily Mail, Quora and the internet generally.
Below are some articles on the more recent history of the cat:
Community outraged after a Rochelle, Illinois Facebook post stated a plan to trap and kill feral cats