Pets are considered to be property in NY, like cars (sorry about talking about your dog this way). To put it into perspective using the car analogy, a friend can take care of your car while you're out of town (and really "make it their own" on an unauthorized road trip), but you still hold the title to the car. If the title doesn't get signed over – or in this case, if your friend can't prove ownership of the dog in a court of law – you retain legal ownership of the property.
Should it become necessary to prove that you are the dog's owner, you can probably produce evidence such as photographs, veterinary records and other friends who would testify on your behalf. Perhaps you have something like a sales receipt from a pet store or a breeder's pedigree, as well.