You get a birth certificate and social security administration registration at birth. The former is issues by the county you are born in, and the latter by the federal SS administration which has a unique number that identifies you.
When you register to vote you register with the county you live in, which will verify that you are an eligable voter in their county using your SSN or state ID (if you have one) and verify it against information the county has in your records about your identity.
You show up the the place you are registered to vote and cast your vote and they check your name off the list of registered voters at that polling place. So to illegally vote you would need to 1) register people who are not registered to vote. 2) Show up to the polling place and vote on their behalf. Given that there will be north of 100,000,000 votes cast in this election this is a horribly inefficient way to commit fraud and there are 31 verified cases of voter fraud out of 1,000,000,000 votes reviewed in a case study from 2 years ago. It does not happen.
There is no requirement in the US to have a national ID card, and lots of poor people and minorities lack photo identification, especially elderly voters who no longer drive or never have. Like my grandmother.
This is not about voter fraud, this is about the GOP trying to prevent people voting because the demographic groups least likely to have the requirements to vote are heavily Democratic voters.
Edit: Citation for 31 credible cases over voter fraud
https://www.washingtonpost.com/news...le-incidents-out-of-one-billion-ballots-cast/