No, we aren't mystic moggies really. but us cat's, well actually pretty much any other mammal or bird other than monkeys and human's have a third eyelid.
Imagine where the normal two sets of eyelids like your own meet vertically, on the inside of these cats have another eye lid that closes horizontally. its a bit of cartilage called the nictitating membrane...eww.
Here you can see the white membranes of the third eye
The purpose of this eyelid is not yet fully understood, it keeps the eye nice and clean with a tear substance that helps clean away dust and bacteria. It's thought that maybe its because cats descend from the African wild cat which is a desert creature, a third eyelid would be really useful with all the sand in a storm.
Once in awhile you may catch a glimpse of our third eyes its an oddity and irregularity cannot be comprehended by humans, its why the Egyptians thought we had a third eye and that we could see things that humans can't. Maybe i can but that's a secret.
If your kitties third eye has stuck around its very important that you rush them down the vets, its a strong indication that they might be really ill. Do not ignore this it may be fatal and the vet will know what we need to get better.
So why don't humans or primates have them? I don't know but that fleshy lump in the corner of your eyes, that's where yours used to be! So I guess you just don't need them anymore, not like me who's always in the tall grass, its good to have my eyes protected!