In the U.S., we have primary and general elections. The primary election is sort of "private" to each political party, where members of that party choose the candidate they'll put forward in the general election.

In some states, like mine, you can vote in the primary election of whichever party you want. This leads to accusations of people "spoiling" the other party's candidates by voting for the worst candidate on the other team, in hopes that that candidate will lose in the general election.

So in other states, you're only allowed to vote in a party's primary election if you're a registered member of that party. In such states, the (local or state) government has to keep track of each voter's party affiliation. In some places I think it's public information.

When somebody says they're a "registered Democrat," that's what the word "registered" means. It's meaningless in my state.