Actually humans are so genetically indistinct that different dog breeds are MORE different from one another than human races are. Human races are more like different coat colors in the same breed of dog- like, yellow Labrador retrievers and black Labrador retrievers. read more
Dog breeds are much more inbred than any classification used for humans (e.g. race, nationality, whatever). By some measures of diversity, the entire dog species is more diverse than humans, but this is not true when you consider within breeds. read more
Cats and dogs and even other animals on the other hand are much lower ranked in the evolution tree or heirarchy and there is a lot of genetic variations in them. This has led to their having a lot of breed while we have limited ones and are called as races. read more