When you register for selective service, you are given a choice between Democrats, Republicans, Independent Free Cities, The Border Cartels, The California Technocratic Alliance, New Furrytopia etc. It doesn't affect your voting choices, but if civil war breaks out, you are automatically drafted to the side you pick.

When you register to vote you pick a party affiliation between Democrat Republican and Independent. This is on the public voter rolls. In some states you can only vote in primaries for the party which you are affiliated. There's a voting strategy both parties apply called crossover voting in these states(especially swing states with this rule) where they promote affiliating with the opposite party in order to manipulate the opposing party's primary.

Your party registration is public, your voting history is not. If you care more about voting in your party's primary elections, register that party. If you care more about your mom looking you up (unlikely), register as her party or "independent".

Lots of people register for one party then change voting behavior as they get older, I originally registered as a democrat.

Many states have you register to receive ballots in primaries. Some states you can choose not to register, but then you are restricted to voting on only 1 ballot. Plenty of people register cross-party to try to sabotage electoral chances by voting for bad candidates in the opposing party primary. I chose not to register so I can vote for either Ds or Rs depending on which slate of candidates is more dramatic and r-slurred (and the Rs are usually very incompetent so they don't even run candidates for lots of local offices and mail out a blank ballot every primary season, so I typically vote for whichever Crystal Mommy D is running)