The traditional medical approach to intersex people was to assign them a sex at birth based on the appearance of external sexual characteristics (basically over a certain size would be a penis, under a certain size a vagina), often talk parents into surgery to support this interpretation and then bring up the child under the mistaken psychological claim that if you bring up a child as a particular sex they identify with that particular sex regardless of biology. This has been historically harmful to many intersex people. More modern approaches are non-interventionist to preserve as many options for the child as he/she grows up, options which include choosing one sex over another or maintaining sexual characteristics that cross binary sex borders. Choosing how to address the child, I.e. as female or male, during this period of development is a difficult question and many families choose to be open with their children about the situation.
This has all come about because of medical discoveries about the importance of chromosomal and endocrinological factors in sex determination and has been aided by the recognition that certain societies place too much emphasis on a binary, strict distinction between sexes. Some cultures, like Native Americans and some Indian sub-continent cultures, celebrate intersex people and don't see the determination of male/female sex as at all important. Intersex discussions are not less about 'fixing' people to conform to what we expect their sexual characteristics to be like, but more about acknowledging who they are and giving them choice to determine who they are.
All this is what makes me annoyed at the "girls have vulvas, boys have penises" mantra, especially coming from a religious source with inevitable connotations that this is how God made us and anyone failing to conform to this standard is somehow wrong.
In that story there are possible parallels with transgender people. Transgender people may or may not have acknowledged biological sources for their desires. Much like other difficult areas like anorexia, we don't know if the desire to change comes from a physiological cause (e.g. chemical imbalance, endocrinological involvement, brain structure), a psychological cause (again there could be physiological reasons or environmental causes), or is simply a desire which it is inappropriate to medicalise. Are transgender people sick and gender reassignment surgery is a possible cure for them, or are they making choices that require medical help to be realized? There is a lot of discussion on this point.
The transgender discussion raises the question what is a woman and equally what is a man? Again this ties into intersex discussions as intersex people present in ways that challenge the binary definition. But forget intersex if you really can't see this point. Imagine a person born biologically female with an unproblematic and functioning female anatomy. She then asks a doctor for a penis transplant (this is a thought experiment, go with it) which is successful so she now has sexual characteristics of both sexes, what is she now, male or female?
She then has her womb and ovaries removed....less female than before?
Now she asks for her vulva to be cosmetically removed so she only has a penis. Can she still call herself a woman?
Personally I don't mind who calls themselves a women, it doesn't upset or offend me if other people see themselves in that way. However, I do think that because women are often minorities, often greatly at risk (from lack of access to basic human rights) the question of who is included in this category can become very contentious.