They identify as a man because gender is defined externally to the individual. They were born, the doctor saw a penis, said, "it's a boy" and those around them reinforced male norms onto them: parents tend to cuddle boys less than girls; enforce stoic principles (men don't cry); encourage rougher, more violent play; and encourage more independence. There's plenty of references for these points so I didn't feel the need to provide any.
Most people don't challenge their identity if they don't need to and changing parts of your identity can be traumatic (ask queer person what coming out was like for them). The crisis of masculinity, as with any cultural crisis, is just a conservative, reaction to something that challenges them.
Gender is cultural and temporal so changes all the time; high heeled shoes used to be worn by rich Persian men, pink was the color that boys wore because pink was thought to be a watered down red, the color of the British army that they would, of course, eventually join. The Male Breadwinner model is an interesting way to frame the idea that the man is provider for the family. Prior to the Industrial Revolution the whole family provided.