When last I checked, that was not the case. It would be a great contribution to Open edX if someone could unify those features, so that the registration form and user profile page displayed and edited the same fields (though the registration page could show fewer fields than the profile). I don’t know if anyone has worked on that recently or not.
As for your other question, I’m not sure. But I think that the birth year is never shown on the user profile - it’s just asked to determine if the user is over 18 or not. I also notice that the birth year is no longer available on the new account settings page at https://account.edx.org/ so maybe it has been changed in recent versions. Perhaps someone else can shed light on this?