The best phrase to describe the classification of dogs and cats is mammals.
Here's Why
* Domain: Eukarya (both have complex cells with a nucleus)
* Kingdom: Animalia (both are multicellular and heterotrophic)
* Phylum: Chordata (both have a backbone)
* Class: Mammalia (both have fur, give birth to live young, and nurse their offspring)
While dogs and cats are in the same class, they belong to different orders, families, and genera.