I don't believe so, but the airline does send your info, including birthdate, to TSA's Secure Flight program - it's how they now often don't need your boarding pass, they just match your name and birthdate to the passenger manifests. This also pops up on their computer screen when they put your ID in the machine (boarding pass info shows up on the boarding pass scanner, which is separate), and TSA makes sure both match (roughly, obviously airlines jam together names and prefixes all the time)

Collisions are likely (I have an uncommon, but not unique, name, and I've been on multiple flights with another person with the same name), so just name and flight aren't enough to positively identify someone, they need something additional (eg birthdate).