To be bailment consideration has to be given. That would be if the original owner paid the other party to take care of the dog for a while. This situation is a volunatry give up, no bailment exists.
Still this is a case of property being withheld without permission, The original owner had the other party take care of the dog and did not relinquish property rights. Title to the property, the dog, still resides with original owner.
What the owner needs is an writ of replevin.