Beatrix appears 65 times at rank 1584 on female pets. The name carries three simultaneous cultural anchors: Beatrix Potter, Queen Beatrix of the Netherlands, and Beatrix Kiddo from Kill Bill. Pet owners are almost certainly drawing on the first, the most animal-connected famous Beatrix in history.
The Beatrix Potter Connection
Beatrix Potter created Peter Rabbit and a universe of animal characters that have defined English-speaking childhood for over a century. Naming a pet Beatrix after the woman who gave us Mrs. Tiggy-Winkle, Jemima Puddle-Duck, and Benjamin Bunny is an appropriate and quietly literary choice. It works especially well on rabbits, Potter's primary subject, though dogs and cats carry it equally well. Compare with Peter in the rabbit-literary name set.
Sound and Breed Fit
Bee-AY-trix has four syllables with a strong terminal consonant that lands crisply. The name suits breeds with a certain bearing: Cavalier King Charles Spaniels, Whippets, and elegant cats. The natural nickname Bea or Bix gives owners flexibility for daily use. The human name lives at /names/beatrix.
The Counter-Reading
Beatrix requires a dog or cat who can carry formal weight. On a chaotic terrier mix, the gap between name and personality becomes the joke. On an elegant, composed animal, the name is simply perfect. Most owners who chose it knew exactly what they were doing.