Phil registers 59 times in the pet data at rank 1726, strongly male. It's a truncated human name — Philip shortened to its single-syllable core — and in the pet context it almost always functions as a deliberate irony: giving a very small or very silly animal the most ordinary, middle-aged man name available.
The Deliberately Mundane Human Name
Phil sits in a specific tradition of pet naming where the comedy comes from the mismatch between animal and name. Gary, Kevin, and Phil are the triumvirate of the genre: names so resolutely ordinary that they produce a small jolt of recognition when attached to a cat or a rabbit. The pet-owner relationship with these names is affectionate and gently ironic rather than disrespectful.
Cultural References
Phil the groundhog (Punxsutawney Phil) is perhaps the most famous animal Phil — a rodent turned annual media event whose name was never meant to be funny but has accumulated that quality over decades. Phil Dunphy from Modern Family extended the name's likeable-bumbling register. Owners drawn to this name are usually aware of at least one of these associations. See the human name context at Phil.
Counter-Reading
Phil is short enough that recall training isn't a concern, but it's also so human that it can produce mild confusion in social situations where a human Phil might also be present. For cat owners specifically, the mundane-name-on-cat genre has become slightly predictable — though execution matters more than originality.