Zeus is the most registered mythological pet name in this catalogue, which surprises nobody. What is worth knowing is the rest of the field, and what each name actually refers to — a tag that says "mythology" and nothing else is a tag you cannot check.
Every name here is recorded against a specific figure: Freya as the Norse goddess of love, Bastet as the Egyptian cat goddess, Fenrir as the wolf of Ragnarok. That is what makes the tag auditable rather than atmospheric.
A note on cats and Egypt
Bastet, Isis and Anubis carry a lot of weight for a small animal, and the registers show people choosing them anyway. Worth saying plainly: nothing in this data supports the idea that a mythological name suits a temperament. It suits a story you want to tell.
Every count on this site is a tally of real registrations: pet licences and shelter intakes published as open data by New York, Seattle, Austin, Montgomery County, Cambridge, Zurich, Basel and three Australian councils. Popularity here is the average share a name holds across those registers, so a name common in a small city is not buried by one common in a large one. Nine of the eleven sources are North American, so read the ordering as naming in those places rather than worldwide.