As a point of clarification, this should be "identifies as a woman". Sex (male/female) is a physical characteristic, gender (man/woman) is a social construct.
More info from transman's first link:
> Sex is assigned at birth, refers to one’s biological status as either male or female, and is associated primarily with physical attributes such as chromosomes, hormone prevalence, and external and internal anatomy. Gender refers to the socially constructed roles, behaviors, activities, and attributes that a given society considers appropriate for boys and men or girls and women. These influence the ways that people act, interact, and feel about themselves. While aspects of biological sex are similar across different cultures, aspects of gender may differ.
If what you said is true (and it isn't) people would only need two questions on forms:
"Are you male or female?"
"Are you a man or a woman?"
This would make no sense and so we can safely ignore it.
Sex and gender are separate concepts, and having separate terms for each is a net positive for the English language IMO.
In the UK current best practice (although it's still changing) would be to have two questions: one asking about gender identity and another asking about gender reassignment.
Someone born as male who identifies as female should need to tick "other", they should be able to tick "female".