Breeders can apply for a Kennel Name (sometimes called an affix) that they then use when naming puppies. For instance, William's breeder had the Kennel Name Arianaharbor. William is Arianaharbor Sweet William. His litter brothers and sisters names also start with Arianaharbor, as did the names of the puppies that she bred from another bitch a couple of years later. So if I spotted a Cocker whose name started with Arianaharbor I'd know it had been bred by William's breeder.
Breeders sometimes breed more than one breed, or perhaps change from breed to another. They keep their Kennel Name regardless of how many or which breeds they breed. Louis' breeder now breeds "Imperial" (miniature) Shi Tzus. So there are several litters of those whose names start with the same Kennel Name that Louis has.
If a breeder doesn't have a Kennel Name they can still register pedigree puppies with the Kennel Club, providing both parents are KC registered and are of the same breed of course! The KC will check their database to ensure that a dog of the same name isn't already registered in the breed. There's a limit to how many letters a name can have and I expect they don't allow rude or suggestive names and brand names or famous names. A friend of mine had a Border Terrier called Barney, but they wouldn't allow her to register him as Barney Rubble. I think he ended up as Barney Rabble