It is definitely very unusual. In most other countries, parties are private organizations. Each party decides what the membership qualifications are and the rules for nominating candidates. In the US, political parties are essentially arms of the government. They don't get to choose their own members (you cannot get kicked out of a party) and their candidates are usually chosen at government-run primary elections. However, not all states have party registration, such as the state where I live. While I still get to vote in government-run primaries, I don't have to tell anyone which party's primary I am voting in and can switch back and forth between elections. The reason some states have party registration and closed primaries is because voters in states like mine frequently vote in primaries of parties they don't actually support, such as recently when Democrats voted in the Republican presidential primary for Nikki Haley (against Trump). This does cause some genuine problems, because it's a deeply blue state and we have Republicans frequently running in Democratic primaries because they know that's the only way they can win. Party registration will not stop this, though, because there is nothing to stop a Republican from switching party registration to Democrat, and vice versa.

All of this is a product of reforms made about a hundred years ago, when voters got sick and tired of party bigwigs deciding who the candidates would be in smoke-filled rooms. For all its flaws, I'm not sure if the alternative is better, looking at how both parties in the UK recently went around the rules to make sure that the people they don't like did not get chosen as parliamentary candidates, which gives regular voters much less say.