Voting is a right as an American citizen. ID's are not a right, and there's hundreds of years of precedent that government cannot require you to have a specific ID to uphold your basic rights. They only have to be able to reasonably determine who you are, so, name and address.

Governments get around this by having things like drivers licenses that most people then elect into because it's required to drive.

But governments could simply not issue you their specific ID, and therefore youre voided of your rights. It always comes back to government power vs individual rights.