How does my state know I'm a registered republican?

When I registered to vote at 18, I could pick a party affiliation. That lets me vote in that party's primary elections. I checked the "republican" box. I am a registered republican - although my party has moved pretty far left of my thinking in the 30 years since.

Voter registration is sent to the state they keep a data base which is supposed to be updated.
I don't get how they have a database yet can't verify it's you casting the vote. Until 2016 primary I had to show my DL after that it became asking for just name address and dob.

I think it's mostly at the primary elections when you declare which party's ticket you want.

Apparently some states ask you to register your party preference. When I've voted in a Primary, when I hit the door they asked me which ballot I wanted. In theory, I could participate in an Operation HIllary scheme.......and vote for somebody with no chance of beating my preferred party candidate.