Camilla is a Latin name with roots in Roman ritual: a camilla was a young woman who assisted at religious ceremonies. It's also the name of a warrior queen in Virgil's Aeneid, which gives it genuine mythological weight. For a female pet, it sits in the elegant, slightly formal register that owners choose when they want a name that ages well.
The Royal Association
Camilla, Queen Consort of the United Kingdom, has given the name an elevated contemporary profile. The royal association has settled considerably since the early 2000s, and the name now reads more as regal than complicated.
The Human Name Crossover
Camilla is a fully functioning human name with consistent American use. That quality appeals to owners of Italian Greyhounds, Whippets, and cats with aristocratic bearing generally.
The Counter-Reading: Four Formal Syllables
Daily use will almost certainly produce Cami or Milla. If you love those nicknames, the full name is a satisfying source. If you specifically want Camilla said in full, be prepared to say it deliberately every time, which becomes its own small ceremony.