Punya does mean to have, but in some Indonesian sentences it is implied that you have said thing. In this case it is implied that you have a few cats, so no need for punya here. Though you can put it and still be marked correct, in Indonesian speak it is not needed as it is already implied.
This sentence shows both "I have a few cats" and "I have few cats" and unless I mixed something up, those sentences are quite different.
I answered "I have several cats" and it was rejected.
Is this sentence closer to "I have several cats" or "I do not have many cats" (like, I have only 3, my grandma has 14)?
"a little" as in "I speak a little Indoesian" means something different to "I have a little cat".
The first sentence refers to a small amount of something, the second sentence refers to something being small. So the Indonesian translations are different.
You would not want to say "I have a small amount of cat".