There are two versions of Morgan; one that is male and one that is female. Only one of these Morgans appear during the events of Awakening, which is determined by the gender of the Avatar. Morgan's gender is always the opposite of the Avatar.
As the Avatar can marry any character of the opposite gender, and Morgan's gender will be the opposite of the Avatar's, Morgan can be the sister of Owain, Inigo, Severa, Cynthia, Kjelle, Laurent, Nah, Brady, Yarne, Gerome, or Noire; the brother of Lucina; or an only child. Morgan's birthday is May 5.