I chose not to put my kennel name on my stud dog that I bought from a breeder friend. Afterall, she did the breeding, I had no hand in it. (He ended up being quite popular). That was just my choice.
I don't insist anyone buying a puppy from me add me as co-owner either (even in name only). Also my choice.
Call it breeders option. There is no wrong or right way.
I've done it all different ways. When I register a dog where I wasn't the breeder, I've sometimes left my kennel name off completely, and sometimes I put it in the name. I've done (breeder kennel) Heybern (dogname) and (breeder kennel) dogname at Heybern.
I had that once. As I remember, we used only the breeder's kennel name, as we wanted to use a fairly long dog name. I have seen some dogs that are registered with three kennel names strung together, nothing else.