Yeah, but in the UK, you don't have to (and can't) register your affiliation. That's a matter between you and the ballot box.

You don't have to in the US, either, but doing so is the only way you get to vote in primaries. And, since I don't know that you know, primaries are the votes that occur for the major parties (I think it's just the Republican and Democratic parties, though it's possible that the Libertarians have one) to nominate their candidate. The voting process is done exactly the same way as voting for president, et al. That is, the primary ballots are not privately organized.