For most of legal history a companion animal was property, divided like a sofa. That sits badly with how families actually experience a pet, and New York law has moved.
The shift
In divorce, New York courts are now directed to consider the best interest of a companion animal rather than treating the question purely as a division of an asset. It is not custody in the sense that applies to children, but it is no longer a simple ownership question either.
What actually carries weight
Less than people assume, and different things. Records of acquisition, who has paid the veterinary bills, licensing, microchip registration, and the practical day-to-day care. Loving the animal more is not the test, and evidence assembled before a dispute is worth considerably more than assertions made during one.
The estate planning side
Separately: asking a relative to take an animal after your death is a wish, not an instrument. Without a properly structured provision, and money attached to it, the arrangement depends on goodwill at exactly the moment goodwill is under strain.
General information about New York law, not legal advice. Outcomes depend on the facts of your situation.