Thank you for your question.
Pets are treated as personal property under the law. Where the pet was a gift to the two of them, they continue to own it jointly regardless of the break in their relationship. If one were to sue the other over ownership rights, the court would consider to what extent each of them had provided for the pet financially, taken care of veterinary care, and cared for the dog's everyday needs, such as feeding, walking, grooming, etc. That you paid for some of its care may not weigh so heavily on the court because you would not be a party to the issue of ownership. However, it may be considered.
Please let me know if you have any follow-up questions.
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