In the UK they match you to the votes roll based on post code as well. They ask you for your address to match you to the voters roll. You cant vote twice. If you pitch up and someone has voted "for you", you can lodge a complaint. I think it will be obvious if there are a large number of these complaints.Like SA, your name needs to be on the voters roll before you're allowed to step up to the polling booth, which means you would've had to register to vote in your area first.
Unlike SA, though, there's no verification to prove you are who you say you are, because like you've correctly stated, there is no requirement for a state-issued ID in the UK, because it apparently infringes on people's rights somehow (they'll refer you to the war, and how people were numbered, etc). Also, they don't mark your thumb or anything as a safety to ensure you don't go back later and vote under someone else's name. It's the UK, after all - the chances of there being more than one John Smith in your borough are freakishly high.
Part of the verification process and audit checks ensures that the number of ballots matches the number of people marked off on the voters roll. Observers from all parties are on site ensure that voters are only marked off on the voters roll if they have been issued with a voting doc. Its only possible to have the number of people marked off on the voters roll to exceed the number of ballots, as people spoil or destroy their ballots.
Im sure the US has similar controls.

