Eventually, maybe, depends how hard you hit stuff and the bumper in question. Depending on the vehicle age and owner there might just not be any fucks around to give (with work vehicles being driven by people making less than $20/hr this is almost guaranteed). Snow doesn't tend to be very damaging because you can't really build a straight up wall with it and the outer layers generally conform to the bumper so it's more like a soft push on all the available surface area. It's just that if the license plate doesn't have anything substantial backing it up it will bend.
There's also a significant materials difference between a bumper (either plastic or metal) and the thin piece of metal a license plate is made out of.
I would imagine there's probably some mental disconnect about how damaging ice could be because "hey, it's just frozen water, it's not like it's concrete or something"
