I don't believe that there's a national voter database (that's a states issue in the USA), nor a citizen database - but the amount of information the Census collects is truly staggering, and they are not very "out and about" about how just much that is - quite effectively, as seen by point 3 in the accepted answer. They (apparently) would like you to believe that they just count you every 10 years, and hope you forget the "replacement for the widely despised census long form" the "American Community Survey" (just as widely despised, but by less people at once - it's the same intrusive questions) that they now send to a smaller sample of addresses every year, but that is really just the tip of the iceberg.
Now, they they make great protestations that the data is "solely for statistical purposes" but the nature and extent of data they do, in fact collect makes that questionable on its face, and despite laws making it illegal to use that data for various purposes, those laws could be changed (though it's unlikely.) The limits of those laws have been abused in 1942 and 2004 (that we know of) by providing finely targeted information on the location of Japanese-Americans and Arab-Americans, respectively. Not quite individuals, but how many of each in a census block or zip code. We know what that data was used for in 1942. In 2004 the unconvincing cover story was that it was "to know which airports needed signs in Arabic" (to which the "not a cover story" answer would be "all of them, the whole point of airports is that people from all over travel through them.")
While looking into this (again) because I was "randomly" (I doubt that) selected for the ACS (again - suspicious in the extreme given the supposed sampling frequency being considerably longer than the ACS has been in existence) I tumbled across papers I'll need to find and link 1 2 3 4 for this answer indicating that the census gets (some) data from literally every W-2, 1099, and 1040 filed every year - including name, address, date of birth, and SSN (they don't need to ask for it - they have it already in their extensive files on you, courtesy of the tax code.) "For statistical purposes" but it's not remotely anonymized in their files, only in data that's "publicly released." One of the linked papers mentions an "administrative records database" of more than 300 million individual entries.
Indeed one of the things that severely triggers me about the ACS is that they make bold claims that it's purely for statistics, followed by immediately asking for name and date of birth (which they already have from tax files, but they want to check up on you...) followed by lots of intrusive questions, many of which they already know the answers to, and a few they (probably) don't. Last time I checked, "purely for statistics" didn't come with a need for names.
If your parents ever claimed you, they know who your parents and siblings are, and they track and link that. Filed married - they track and link that. Every address that any W-2, 1099 or 1040 was mailed to or filed from - they have that. It's frankly creep-tastic. To that, they add the smattering of additional data they collect on any census where you were counted, whether as a minor (more family data linked) or a respondent, and any ACS you filled out in the ACS era. You don't need to file taxes to show up this way - any interest or dividend payment over $10 generates a 1099, and any legitimate work generates a W-2, even if you didn't make enough to file taxes.
Aside from the "time since they last 'randomly' picked me" being considerably shorter than the claimed "45 years per household" my main reason to think that the selection is non-random (which would indicate a disregard for valid random sampling) is that we filed taxes from a new address. And while they have records on each person going back to whenever their parents first claimed them on a tax form, here was an address that they didn't have an extensive dossier on with such "vital to statistics" information as toilet status and electrical bills.