My cat communicates in an oddly human way

I have experience with cats. I have been obsessed with cats since "The Lion King" came out, and grew up with two of them. A few months ago however, I just got my first cat for myself (as an adult), and she has a quirk that I have never seen or heard of in a cat before.

When she wants something, she has an oddly human way of expressing it. She mews while doing this head swoop, looking from me to the thing she wants (like the cupboard where her food is, or the room she wants to go inside), and then back to me. It's like something an animal in a cartoon Disney movie would do.

Most strange (and adorable) is her reaction to my apartment's bat problem. (SIDE NOTE: the landlord knows, he's just taking his sweet time sending someone to bat-proof my house). Anyway, a bat got into the apartment through a storage room, and my cat was having fun chasing it while it flew above her in circles; after I got rid of the bat, she mewed pitifully while circling her head around the ceiling, seemingly begging to have the bat back. Maybe a week later a second bat got in, and the same thing happened (but I got rid of it more quickly this time.) She made the same gesture. And a day or two after that, she was still begging me to let her into the storage room where the bats were coming from, circling her head at the ceiling.

Is there some rational explanation for this gesture? I know cats have good memories, so I can believe her remembering where her food comes from and what chasing the bat was like. But by god, I have never seen a cat use such oddly human gesture to express her wants before. Does anyone else have a cat that does this?