okay, by definition - grease is rendered (melted down) animal fat; when grease solidifies, voila, fat!

Ah hah! We'd call (food) grease "oil", then.

Surely you don't fry things in "cooking grease"??

(or "olive grease", "corn grease", "groundnut/peanut grease"...)

No of course not, once its solid its fat (and yes you can by chicken fat in a little package in NY grocery stores.)
but olive and peanut never solidify-- they are oil. Olive oil, corn oil, and you would grease the pan with butter or lard before cooking (say a cake). and other instructions might say "drain of all the grease/pan dripping except 2 tablespoons full, add flour... for making a roux.

Vegetable oils advertize that food cooked in their oil doesn't taste greasy.

in a pinch, i have used "crisco" a brand of solid vegetable shortening (solid oil!) as a grease. the main bearing failed on my concrete mixer in the middle of job, and a improvised with a large brass washer and some crisco!
i did mention i was--past tense--once married?.... Not any more.