That's what we do in the US. You are assigned a polling location based on your home address. You can't vote anywhere else. If you try, they turn you away.
You can do a provisional ballot (for people who recently moved, and poll data isn't updated, etc) and they validate your ID/address/etc later.
In the United States, it hasn't been. The article you link to doesn't even mention the United States. To do it on a large scale requires cooperation from the people running the election, and the US isn't (yet) that corrupt.
The US system isn't completely robust against it, and perhaps some day it will be a problem. But right now there is no evidence that it is a problem, and all of the attempts to "fix" it are clearly aimed at preventing some people from voting.
You can try voting again at other stations, especially since they don't require ID. You just need the name of somebody assigned to that station, who hasn't already voted. There is a signature check if there is a suspicion, but that's rarely done.
But that's practically never done. The risks are too high, and to have a significant impact would require enough votes to make it certain you'd get caught.
For what? In my state there's no requirement to show ID. When I first moved here I attempted to show mine at the poll and the poll worker told me to quickly put that away and she didn't want to see it. I'm not even sure it's legal for them to ask for ID here, given her panicked reaction to me trying to show it.
Since then I've voted in this state for around 10 years and it's always the same. I could say I'm whoever I want, and just be given a ballot.
Edit: I don't live in NY either, as the other poster used as an example. ID should be an obvious and necessary requirement, but it isn't in many states.
And again, you still have to be willing to commit a felony to move the needly by ONE vote, which is not likely to be very common. The risk/reward simply isn't there.