That is the documentation they ask for in the application. It's enough for them to understand who you claim to be. They then consult their own records to establish if that identity is eligible to vote. Then finally, on Election Day, you show you are that person.

At that last part, Election Day identification, is not even that important, since the same person can't vote twice. So if you impersonate another person that will be quickly detected. It's not a useful strategy to alter the outcome of an election.