When the general election comes around, your ballot is still secret. So you can publicly declare a registration for the Republican party, but then vote secretly for the Democratic candidate. (Or vice versa)
In the UK, if you want to have some effect on which candidates are selected by a party, you usually have to join the party and go to meetings and stuff. In the US, parties mostly use public primary elections to select candidates, and the primary elections are run by the same government bodies that run the general election. That’s why the voter registration cares about the party.
When the general election comes around, your ballot is still secret. So you can publicly declare a registration for the Republican party, but then vote secretly for the Democratic candidate. (Or vice versa)
In the UK, if you want to have some effect on which candidates are selected by a party, you usually have to join the party and go to meetings and stuff. In the US, parties mostly use public primary elections to select candidates, and the primary elections are run by the same government bodies that run the general election. That’s why the voter registration cares about the party.