Maybe.
If you settle your divorce by agreement, part of that agreement could be to share custody of a pet.* If you take your case to trial, a judge would be required to award the pet to one party or the other.*
Pets are considered property, so the agreement would be some sort of joint ownership of the pet with contractual obligations. It would not be “custody” in the sense that the court maintains jurisdiction over the custody of children, with the ability to use the court’s contempt power to enforce.