This sounded weird to me, since I'm used to the situation in which organizations (such as parties) keep their own membership record; and voting in inter-party elections is based on that registration and has nothing to do with state authorities.
Rules differ by state in the United States (US), but in general, the state runs primaries which choose the two major party candidates. These are publicly financed. They may or may not be limited to people who are registered to vote as a member of that party (if the state even has party registration, not every state does). If not limited to one party, sometimes it is limited from the other party.
The major parties (Republicans and Democrats) are privileged in the United States. The winners of their primaries (first round) automatically get a position in the general election (second round). Most other parties have to pay for their own candidate selection process and have to get each candidate listed manually. There generally is a process by which they get signatures. Enough valid signatures and they get ballot access.
In addition to that, people can run for party positions. These often occur as part of larger elections, run by the state. Often there will only be one candidate for such a position, so the "election" is somewhat moot. Other times, these are picked by caucuses. Caucuses may be run by the parties, although there are also caucuses that substitute for primaries and are state funded (e.g. Iowa).
Most of this was originally a combination of cost-saving and anti-corruption measures with the convenient side-effect that it reinforced the two-party hegemony.
It has also produced a system where the two major parties are part of the system. They can't pick their nominees freely because each state has rules about how nominees are chosen. Because the state rules control ballot access, they also control the party process.
In theory, a major party could organize itself as a purely private entity and control its nomination process. But then the party would lose its automatic ballot access. It would become like the Libertarians and the Greens, where it would have to gain ballot access by collecting signatures. In districts (congressional or statewide for Senators and presidential candidates) that did not collect enough signatures, the candidate would not appear on the ballot.
As a practical matter, both major parties prefer their privileged status to having the freedom to choose. This may change, as in the last primary season, one party was dominated by an independent who had been a reality television star while the other saw a strong insurgency from an independent who made a career out of being a gadfly. Perhaps this will lead to reform of the process.
More specifically, what different duties and privileges do you have as a "registered Democrat"/"registered Republican" as opposed to a member of the state or the national Democratic or Republication party?
Registration is an official action. The effect varies by state. In a closed primary state, people can only vote in primaries if their party registration matches the primary. In some states, independents can choose a primary in which to vote but Democrats and Republicans are restricted. Other states are open primaries, where registration doesn't matter.
In New York, voters are limited to just their own party's primary, but a candidate can run under multiple party lines and in the general election the party results are combined (fusion). In California, party registration only matters for president (other primaries are non-partisan), but the Republican party primary is closed while the Democratic primary is open. In both states, the primaries are state run affairs and limit what candidates can appear in the general election (except for president, which has its own goofy rules).
Kentucky, Louisiana, and West Virginia are generally considered Republican states. They voted overwhelmingly for Donald Trump for president (and previously Romney, McCain, and Bush). They have mostly Republican congressional delegations. But they are majority Democrat in registration (technically Louisiana is only a plurality, but with a 19% lead).
"Member" is an amorphous term. It could mean party registration. Or it could mean donations. Or it could mean that they answered a poll saying that they preferred a particular party. Or it could mean that they have a formal party position, like a delegate. I've seen it used in each of these ways. Without additional context, "member" doesn't tell you anything.
Note that when people talk about party registration in the US, it is registration with the state. The two major parties in the US are not private organizations, free to choose however they want. They are officially recognized parts of how the system works. Each state sets its own rules for elections and such elections control the party process as well.
The parties could give up that status. But if they did, they would lose the funding that went with it. They could choose their own nominees but they would have to pay for the process for doing so. And they'd have to get their nominees on the ballot through their own actions. They choose not to do so.
The two major parties in the US are not private organizations. They are quasi-governmental organizations (contested). They enjoy privileges not available to purely private organizations. For example, their nominee selection is state funded. And they have to accept the rules set by the individual states that come with that funding. For example, the party nominee is the person selected by a plurality of those voting in the party primary or through a state approved caucus.
Some would like to argue that "quasi-governmental organizations" is not the right description. But the fact remains that even if the parties can choose their nominees by a process other than the person selected by voters, it doesn't matter. The person who appears on the ballot as the party's nominee will still be the person selected by voters. Because that's how the state determines ballot access. And ballot access is naturally within the purview of the states unless they delegate to the parties.
The "party" primary is a state election. Note how Louisiana, Washington, and California don't have party primaries. Their processes just select the top two, possibly of the same party. Presumably the parties could have separate nomination processes there, although they have not yet chosen to do so.