They both belong to the order Carnivora. A family is a stable, robust taxon in the Linnean system. In families, we can easily recognize a large number of shared characters. In fact, we recognize more in this group than in orders or genera. Classes are being wiped out by cladists because they are paraphyletic, but families will probably survive.
Character analysis suggests that they arose from another dog-type critter.
We can use morphological changes to separate organisms at the family level, but it is hard to tell which organism is the most recently evolved type. Let's look at some characters we could use to compare our animals. We could look at face shape, ear shape, ability of claws to retract Dental formulas of cats and dogs are different, but it is hard to use this trait to compare dog-like critters to each other. This is where molecular analysis comes in. Molecular analysis is quite useful because DNA doesn't lie.