According to the answers given here, you'd normally use Katze instead of Kater in normal conversation, even for a male cat. The only time you'd use Kater is a) if you're a veterinarian or breeder who has a professional interest in the sex of the cat, or b) if you want to specify the sex for some other reason, e.g. Ich habe sowohl eine Katze als auch einen Kater. -- "I have both a female and male cat." This is more or less as I expected, constantly calling your pet cat a tomcat would be a bit weird in English and I assumed the same would be true in German.
The problem is that I'm watching a television show (Bosch) with German subtitles, and they use Kater more than once where it seems like Katze would be the normal word to use. In the first scene, a man is being arrested and wants to make sure his cat will be taken care of:
Ich kümmere mich um Buddy, Sir.
In the second scene the man's daughter comes to collect the cat and the same detective want's to know what she's doing in the man's house, since she's poking around as well:
Was tun Sie hier?
Ich wollte den Kater holen.
Ich hab mich nur umgesehen.
So my question is, since nobody, least of all the detective, cares about the sex of cat, why use Kater both times? My thinking now is that it's some quirk of the translator. The German audio uses Kater both times as well though.
The reason I'm curious about this is that one of the first things you learn about German as an English speaker is that German uses grammatical gender instead of natural gender as in English. So even if you're talking about a male cat you might say:
Wo ist die Katze? Sie liegt auf dem Sofa.
Specifically, you'd use sie instead of er even though the cat is male. But if you use Kater instead of Katze for no good reason it kind calls the whole grammatical gender theory into question.