In general, your registered party is part of your state voter registration. Some states use that to determine which primary you can vote in (“closed primary”), while others let you choose a party at the time of voting (“open primary”). Either way, though, voting in both parties’ primaries is operated by the state as part of a unified process, at the same location; so it shouldn’t be possible to vote more than once.

If you’re in a caucus state, that process might be different; I’m not sure.

You 'register' as one, the other, or something different (L, Green, etc.) typically at the DMV or something like it. The selection there is a radio button, not a checkbox.

Of course, nothing prevents you from being a member of whatever party you want, though there are sometimes laws prohibiting when you can change your registration. Oftentimes, you're prohibited from changing your registered party affiliation within a month or so of the primaries.

Your voter registration record says which party you're in. The election official would likely make some inquiries if yours said "all of them" on it.

State dependent. In Texas, you don’t register for a party. You just vote in the primary you want (but you can’t vote in both primaries.)

Shouldn’t who you vote for be confidential and you can change your mind till the day? And why can’t a person vote one way for local and another for state ? I don’t get the system

The party registration isn't counted as a vote for that party - when you show up on election day you can vote for whoever you want.

What it affects is which parties' primaries you can vote in.

The primaries aren't official US elections. They're unofficial elections set up by each political party (which are private, non-government entities).

You can vote however you want in official elections.

I would be careful about asserting that. In PA there can be referendums on the primary ballots. And the results can be binding (sore loser laws, etc.). They're pretty official looking around here.

This definitely differs between countries. Where I'm from, I'm a registered member of all political parties. No rules against it.

You can't be registered for multiple parties.