Family packs were matriarchal and dogs took the surnames of their mothers. Packs that formed outside of the family took a male for their leader.
Breeds, as you call them, didn’t exist. A dog with a breed for a surname may have resembled that breed and they may not have. For example, Brody Beagle was only part beagle. A pack of long-haired white dogs lived over the mountains, but even they had physical differences. I imagine there were once packs of identical looking dogs, but not during my time.
You can learn more about the dogs of Barkingburgh here, A Bit About Dogs.