This is not a good explanation, because you could just as well say "I have cat and two dogs". The fact remains that "a cat", "cat", and "the cat" (and "one cat"

This is not a good explanation, because you could just as well say "I have cat and two dogs". The fact remains that "a cat", "cat", and "the cat" (and "one cat") all have separate meanings; while for plural, you only have "cats" and "the cats".