Post-election audits routinely show that there are few, if any, non-citizens voting. I'm not sure how the states check registrations between when they're filed and election day, but it's a crime so it would be a dumb thing for a non-citizen to do (particularly someone who isn't a permanent resident and at risk of deportation).

If you pass a law saying that you have to provide a passport or a birth certificate to vote, you will be disenfranchising millions of native born Americans. The burden to register to vote will be enormous for a lot of people. Getting a passport takes months and tracking down your birth certificate or getting a new one could be similar. This isn't the one grandma who can't get a driver's license. That's what Johnson and Trump proposed. And if we're talking about doing this fairly, every single person should have to re-register because they didn't have to prove it the first time.

As far as your gun analogy, in Alabama you only need a driver's license. Much easier than proving citizenship and the same thing you would need to register or vote.