I showed it to my boss, in case it represented a hazardous level of solvents in the air. (30+ years' experience vs. my not quite 20.)

He started peeling it (damaged already, I didn't care), and it's some kind of vinyl applique. He said it could be airborne solvents (it wouldn't take much, thin as this is) or it could be just moisture loosening it. Since I fill it with ice water each morning, moisture's a definite possibility.

The rest has been scraped off with a card, and it's nice and clean.

Thanks for the input, all!