It's weirder still because you often don't actually have to register with a party, even to vote in the primaries like other commenters have mentioned; it depends on which state and which party. [0]

For example, I just voted in the Minnesota primaries. Ours are open primaries, so you don't need to register with any party and the ballot has every party on it. Though, it spoils your ballot to vote in multiple parties' primaries. So you gotta choose, but you do that privately.

[0] https://ballotpedia.org/Primary_election_types_by_state