You can register as “unaffiliated”, but depending on your state what options you have for voting in primaries depends on your affiliation.

Some states only allow you to vote in the primary for your affiliated party, others let you pick one or the other regardless, some will only let you vote in one parties if you are affiliated but the other has no such restriction. The system is a giant clusterfuck of rules, but everyone just kind of works with it for now.

Your affiliation is part of your voter registration, so it’s held by your Secretary of State’s office.