Kočka and liška are generally used as a blanket expression for cats and foxes. BUT at the same time are the female version of both. (The opera Příhody lišky Bystroušky by Leoš Janáček is translated as Cunning Little Vixen.)
If you have a need to point out that the animal you are talking about is a male, it would be KOCOUR for a tomcat and LIŠÁK for a male fox.
It is somewhat strange and some may object it is treating the cat as a human being, but we accept it. Some people will say it like that (children, pet cat owners...). There was a rant from a Christian politician against a certain lady that used "umřít" (to die) about her dog instead of "chcípnout". Such people may object to your sentence.
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I've gotten this question a few times this lesson and I was curious about word order in this case. The only order the program accepted was the one listed above, but I would have guessed that "Ptaka zere ta kocka" would be dramatically accepted as well, even if it emphasises that it is a cat eating the bird and not some other animal. Am I missing something?
Th word order in Czech is not random. It is governed by the topic and comment structure https://en.wikipedia.org/wiki/Topic_and_comment It most often changes the meaning or more exactly the focus of the question. The new information typically comes last. If you begin the sentence with the bird ("Ptáka...") it really makes much more sense for THE bird and not for A bird.