Earlier quoted context omitted.
Birthday is almost certainly public. You can get it easily from the DMV or from any of several dozen commercial providers that resell government data: https://www.dmv.org/public-records/ That it's used to confirm identity shows how weak identity-theft protections are at most institutions, not what's public information. (For that matter, mother's maiden name is basically public information as well: you can get it from…
The site you linked to is not affiliated with any state DMV. It links to a sketchy background check service that appears to be a scam. Edit: spelling
There's no perfect identifier. That's why we need security. This is a major lapse.
Pretty sneaky of the comment above yours to try to pass dmv.org as a government website.