You have to go to Account > Settings (the gear icon) > Manage Profile; then under Personal Information (with your address and such), click Edit. Annoyingly, it seems to make you verify your identity every time you do this (i.e. it'll text or email you a temporary code you have to enter). Then you can see your complete personal information, including birth date.

Oddly enough, unlike on the web interface, it seems to allow you to modify it on the app. I didn't try doing it, though, so I have no idea if it would actually let you. Since the web interface doesn't let you, I'd be surprised if it would actually go through.

EDIT/UPDATE: I just checked again in Chrome - and it actually displays as the correct day now! No idea why it wouldn't have updated before (even after it did update when viewed elsewhere), but it seems it was probably wrong in Marriott's system after all. Going to edit my previous post to reflect that.