Story wise, what does a birthday say of a character? Unless your book is about of astrology, or it is a special kind of date, seems pointless to me. It is only a bunch of random numbers. You may as well give a SSN.
Yeah, I know most of the data given in descriptions of people and places is mostly irrelevant anyway.
Pretty sure you can build great stories around a birthdate, but I don't think they are interesting by themselves.