Didn't read all the posts so I'm not sure if anyone explained already, but:
You can clearly calculate their age by knowing at the year their mother died in childbirth: 1159. The game begins in 1180. So Byleth is 21 at the start of the game.
Now, Byleth's exact birthdate is a little strange because despite the game letting you choose one, according to Jeralt's diary it sounds like Byleth's mom died in childbirth during Horsebow moon, or at least just before it.
This could mean that the in-game birthday is just a made up date that perhaps Jeralt gave Byleth as their new birthday or something. But the fact that Sothis claims to have the same birthdate is very strange then.
I'm scared that all of this is explained in a route I haven't played yet, haha.