Registering to vote has been made so easy it can be done by accident. Then months or years later when an election is coming up they'll get a voter card on the mail and think that means they can vote even though they're not legally allowed to vote.
I'm kind of incredulous at this if I'm being honest. How can you register to vote by accident? Every form I've seen a copy of asks if you're a citizen and gives you a warning about that. Do you have a copy of the form or screenshot you're referring to that makes this easy to do by accident?
I don't, but it's easy to find people who realized this happened to them, getting scared about their immigration status and/or breaking the law. And the DMV isn't the only way this can happen: