If there is any kind of personal relation to a cat, like when it is your pet or you are acquainted enough to know the name by which it is called and doesn't come, you'd usually refer to it gender-specifically as "Katze" or "Kater". An exception may be if you are referring the cat to a stranger, like "haben Sie eine schwarz-weiße Katze gesehen?". If you are relating specific feats, like its ability to catch rats, you'd usually revert to the gender-specific form as that makes it personal.
If there is a multitude of cats on the premises that is kept to keep rodent populations in check (farms, stables, and so on), the likelihood of individual names goes down along with the likelihood of gender-specific references.
As a corollary, "die Kater" is almost exclusively used when referring to the male subset of specific cats and the non-deterministic plural "Kater" is basically only used when referring to behavior specific to male cats. "Die Katzen" may include male cats as does "Katzen".
While "Haben Sie meinen Kater gesehen?" may be asked of a stranger when looking for your male cat, "Haben Sie meine Kater gesehen?" would be quite unusual even if both cats you are looking for happened to be male.