In practice this doesn't happen, I just assume because it's easy to detect, it's a felony that comes with a prison sentence (and a lot of people would be looking for you) and because it actually creates a lot of problems for whichever candidate receives the votes.
I think an alternate scenario, where ballots are sent to a lot of people with the voter's name and address information "helpfully" prepopulated on the form, incorrectly, is perhaps slightly more of a threat. In this case people are at risk of casting a ballot that will be rejected and their board of elections might or might not notify them there's a problem. Sending prepopulated ballots is illegal where I live because an organization trying to be helpful in this way could inadvertently disenfranchise people by mistake with a bad list, but there's the other risk as well.
Again, easy to detect and a crime, and I don't think it's been a problem here.
The security features of the process do not depend on mail-in ballots being unique or scarce.