Huh. My Rx labels just peel off. When the pharmacist puts it on crooked, it bugs me. I peel it off and re-stick it straight. Same with the veterinary Rx labels I use for work, if I manage a crooked one, I peel and re-stick.
I’ve lived in neighborhoods where people habitually went through the local trash dumpster. Mostly, it was old guys or the homeless looking for cans or other stuff to sell for a bit of cash, but according to the local police there were another sort (which might or might not overlap with the first) that would use any information gained for nefarious purposes - thus, identity theft from mail/bills with personal information, and if they found out there were narcotics used by a person at a specific address they might break in looking for the drugs.
Which is why you shred mail and other documents with personal information on it.
A bottle label for something like penicillin is pretty innocuous. A label for some other drugs, like painkillers, could potentially make you a target. It’s not so much having a dedicated enemy as low-level scavenger-type criminals looking for an opportunity.