Yes, a dog can have two owners. In fact, many dogs have multiple owners. For example, service dogs often have multiple handlers during their working life. Each handler becomes a primary owner to the dog while they are working together.
A breeder contract is a legally binding document that sets out the terms and conditions of a puppy's sale. It details both the breeder's and the buyer's responsibilities and expectations, providing a clear framework for the dog's care and management.
In the painful event of a serious dispute, a court is more likely to enforce terms that are clearly spelled out as essential to the Breeder's agreement to sell the puppy in the first place. Not everything is a Deal Breaker - Remember, if the contract is too overbearing and unreasonable, it likely won't be enforceable.
It may well depend upon what country you're in. In the US the short answer is no, they can't take the pup back or force you to alter the pup. Under US law, dogs are property, so if you bought it outright (no co-ownership arrangement) it's yours to do with as you see fit.
Co-owning means that there's going to be a contract, which is going to specify ALOT, and it also means that at various points "your dog" is going to be in the direct control of a person who has just as much legal right to the dog as you do, which means that you're not going to get her to change her ways.
A Binding Agreement: Remember that a breeder's contract is legally binding and outlines the responsibilities of both parties for the dog's care. Fully understand and agree with these terms before committing.
Co-ownership is when two or more people have their names as owners on a dog's AKC registration papers.
While AKC registration application forms provide for more than one owner, we recommend that co-ownerships be avoided. It has been our experience that purchasing a dog “outright” is preferable.
The co-owner keeps the puppy and is responsible for day-to-day food and vet bills, and entry fees for any sports they want to do, but the breeder covers any breeding related expenses, health testing, and costs of showing if they choose to show in conformation but the co owner isn't interested.