That genetics aren't as big a deal with identity politics as they are cracked up to be.
 
You just said it yourself, you don't need male or female sex organs to determine gender. Gender is just the way that we 
culturally express the characteristics associated with primary and secondary biological sex. Primary sex refers to reproductive organs, secondary sex refers to the traits that are used to distinguish the biological sexes that are not directly related to reproductive organs.
 
While primary sex characteristics are biological and can't be changed completely, secondary sex characteristics are entirely changeable, and are not absolutely connected to primary sex traits... there are 
hairy, deep voiced women, etc. 
 
So really while these people cannot change their primary sex characteristics, they can change their secondary sex characteristics and the social/cultural associations that come with them.