Already under the law or registration and voting process, an ID like your social security last 4 digits, or your drivers licence number are required on the registration application....The point remaining that somewhere along the line, you have to produce an ID to exercise your Constitutional right.
Then the govt office vets you just from that info....for approval or not.
Then, under federal law, the first time a newly registered voter votes.... he or she has to show identification that they are the person registered to that address registered on file....this can be their most recent electric bill with name and address or driver's licence....etc etc.