It's not actually part of our electoral system, though you'd never guess it from coverage nor results. Neither the Republican nor Democratic parties in the US are actual federal agencies or anything. They're technically "private clubs", except that at this point they function like psuedo-government entities in most respects.

Our two "major" parties have an in-party vote ("primary") to determine which single person from that party will run in the actual, legal election ("general election"). Some state parties only allow people registered with that party to vote in their primary, since they're non-government institutions and can set whatever rules they want.

Technically, people aren't entitled to vote in primaries at all. Either party can declare that anyone is their nominee without a vote the same way that you can decide you're running for President if you meet all the qualifications. You don't have to ask anybody. It's just that you're not taken seriously unless you're "anointed" by a party that matters.

So, yeah, our system is broken, but it's not that broken. You can vote for whomever you'd like in the actual election regardless of which, if any, party you're registered with. You could in theory be registered as a Republican and consistently vote Democrat in every general election, no one would ever know.