There are no "pet custody" laws. The reason is that custody applies to humans. Pets are personal property, and the laws that relate to any other property (e.g., cars, furnishings, clothing, jewelry, etc.), apply with equal force to domestic animals (pets).
There are, of course, laws about cruelty to animals, and dog bites, service animals, etc. But, they are still property, and control over the animals generally devolves to a question of property rights.
If you purchased a dog, and someone else is in possession
of the animal, then you can go retrieve the animal, and if that person refuses, you can call the police/sheriff and report the matter as a theft.
If the other person claims that you gave them the dog, or something similar, then you may have to take the person to small claims
court and sue for the value of the pet. But, first you need to try to recover the dog in an amicable manner -- before starting down any legal road.
if this doesn't make sense to you in any manner, please feel free to ask for clarification, and I will be happy to try to assist you further.