The trouble with lists of unique pet names is that most of them are invented. A name nobody has ever used is not unique, it is untested: you find out at the vet, at the groomer and in the park whether people can hear it, spell it and say it back.
Every name here is rare and real. Each appears in the public registers this site counts, so a live animal answers to it — and each holds a small enough share of those registers that you are unlikely to meet another.
Rare and short at once
Rarity usually costs you syllables. These are the exceptions — four letters or fewer, and still uncommon.
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.