Some conclude that if someone is born with only a male sex organ but is strongly confused about his gender (or strongly thinks he is female) then he might really be female.The logical form of this reasoning, however, is completely fallacious.
Let me preface my comments by saying that I'm not an expert in this matter; nonetheless, I'll do my best to think clearly about the issues.
I'll take your questions one at a time (I've re-worded some).
It is perhaps best to start by making the distinction between gender and sex.
Some think that one's sex determines one's gender, while others think gender is completely "socially constructed," so that differences between the male sex and the female sex are nothing different than one's sexual organs.
I think you'll find his response below to be both interesting and illuminating.