When you describe a cat or dog's personality, what's the noun for it? "My cat has this personality of ..." just sounds wrong.
Veterinary behaviorists would call this temperament.
Saunders Comprehensive Veterinary Dictionary, 2nd edition, defines it as:
the peculiar behavioral character and mental cast of an animal.
"Personality" is the right word. Also, they can be personable, and you can impersonate animals too.