I think its mainly because there are more purebred dogs than there are cats, and more distinct and thus recognizable breeds of dog. Breeding cats in the US isn't as popular, for whatever reasons, as breeding dogs, and so we have more "purebred" dogs.
Also, there's such variation between different breeds of dogs--think of the differences between chihuahuas and great danes, for example, so its easier to recognize specific breed traits in dogs than in cats, where many breeds are very similar in size and shape. The only breed trait I can really easily recognize in a "mixed breed" or DSH/DLH breed of cat is the Persian "flat-faced" tendency, which you do see every now and then in a "shelter" kitty.
Though my two kits are both shelter kits and true DSHs, I see physical differences in the two of them which point, I think, to evidence of different breeds somewhere in the distant past of their genetic history.