Actually, to correct a misconception I've seen on this thread -- people under 18 can legally co-own a dog (I've never heard of the term "co-dog" before PT though). Many juniors "sign onto" (which is a legal documentation process of simply signing onto the dog's papers as an owner) dogs to show in junior showmanship.

And people under 18 can legally own a dog as well, but I don't think they can legally sign contracts, which is why most dogs are simply under the parents' names (breeder contracts usually need to be signed when the dog comes from a responsible breeder).