amelia, being born with no penis doesn't automatically make someone transgender. That sort of person would be a biological male born but just one without a penis due to developmental abnormality.
Or possibly the word you're looking for in that case would be intersex, often babies can be born with BOTH types of genitals and the doctors have to work out what to do and make a decision about what gender to 'give' the child. Sometimes they mess up and cause a lot of heartache when the child is raised as one gender but emotionally feels closer to the other, which can cause conflict later in life if they don't tell the child what happened at birth and hide it from them.
Some people are comfortable being born intersex and staying relatively androgynous, identifying with both genders without feeling the need to pick one. I knew one such person, I saw her as an ordinary young teen girl but later discovered she was 'both' for want of a better word, she was on the channel 4 doc 'Intersex' and that's how I found out. A really remarkable person who accepted both of her dual genders.
But transgender generally means being born with one of the other and then later in life, sometimes as a child, teen or adult, choosing to transition the other way. Obvious it is a grey area as intersex and trans people get lumped together, for good reason because the issues are similar, but they not the same.
Just thought I'd clear up that point. :)