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My experience is that it’s pretty fragmented with different agencies or programs tracking information separately. You obviously need to let the DoL know where you’re living as part of registering for whatever, but they don’t share that information with the unemployment people or whoever. And that’s before you get into the state vs federal divide.
Your SSN is often used as a federal registration number even though the card has “do not use for identification” on it in great big letters. Most functions just trust state ID for authentication purposes and use SSN as a label. An identifier in the database sense rather than the authentication sense. At least in theory.
See also how so many of the laws governing this are frankly archaic at this stage, with congress to busy fighting over whether the government should exist or not to actually govern anything effectively. (Note: government inefficiency has never been treated as a reason to govern better, only to govern less and assign more functions to for-profit private entities.