If you are trying to express the fact that you do not have any cats, then the correct English would be "I have zero cats", or even better: "I have no cats". Alternatively, you could use the singular here and say, "I have no cat." It is incorrect to say "I got zero cat", because cats are denumerable. If you say, "I got zero cats", then you are talking about a hypothetical acquisition of cats, in which you did not succeed. It is incorrect to say "I got zero cats" if you really mean that you do not presently have any cats. You will often hear "I've got no cats". That is not quite as correct as some other suggestions, but it is colloquial.
With language, most of the time, the answer to why anything is the way it is is simply convention. Why is English spelling the mess that it is? Convention. Why are there so many exceptions to rules in English? Convention.
Curiously, in Latin, the adjective 'nullus/nulla/nullum', which means 'no', declines to follow the noun it modifies in gender, number, and case. So, in Latin, the sentence "I have no cats" would read "Nullae feles habeo". Or "I have no cat" would read "Nulla felis habeo". There is no 'zero' in Latin, so I can't show you how that (doesn't) work.