The US was organized around a very limited federal government (this has changed over time). Each state is responsible for elections within the state. There's no exact national registry of citizens, although Social Security keeps a de facto registry as most citizens are enumerated by that administration and their records include citizenship indication. Strictly speaking, registration with social security is not mandatory, it is required for employees and a child must be registered to qualify for tax credits and it makes administrative tasks simpler.
Within each state, voting is typically administered by county (or similar level) offices; although I think many states have a state wide process to register.
A given voter will vote for many positions (and often some propositions), and electoral boundaries for positions don't necessarily align with anything else or each other. So there's a lot of complexity there.
The residence address of the voter is the important thing that decides which elections they can participate in, but since there is no real registry of residence (de facto or de jure), it needs to be collected seperately.
[1] https://www.service-public.gouv.fr/particuliers/vosdroits/R1...