It's highly counterintuitive, but the sex you develop as has nothing to do with whether you have a Y chromosome or not. Instead it depends on whether there's androgenic hormones (testosterone and DHT) present during your prenatal development.
In normal male development, the presence of a Y chromosome causes the fetus to develop testicles, which then promptly start churning out the hormones required for male development. However, it's the hormones and not the Y chromosome that cause the fetus to develop as male, and if anything prevents the hormones from being produced or doing their job, then the fetus will instead develop as female instead of male. There's two medical conditions which prove this to be the case: Swyer's Syndrome, and Complete Androgen Insensitivity Syndrome (CAIS). In Swyers Syndrome, the testicles fail to form and so no androgenic hormones are produced; in CAIS, the testicles form and produce their hormones as normal, however a mutation to the gene for the androgen receptor means that the cells throughout that person's body are completely unable to detect or react to androgenic hormones, so all their development takes place as if those hormones weren't there. Either way, a person who is genetically male (46,XY karyotype) develops as female, and people with these conditions look just like ordinary women, so much so that often the condition isn't even picked up until puberty, when menstruation fails to start.
CAIS is a particularly good example proving how the whole process of sexual development depends entirely on hormones, since, as rare DSDs go, it's relatively common (there are thousands of CAIS women alive today), so people with the condition have been well studied. Also, the only difference between these people and the genetically male people who develop as male is that their androgen receptors don't work (or are missing altogether). Everything else, including hormones, enzymes, Y chromosome and other genes, is unaffected. That basically proves that, in humans, male development is entirely driven through androgen receptors (and, without androgen receptor activation, female development will occur instead).
The really important thing from our point of view is that CAIS women seem to behave exactly like ordinary women, and seem to be universally happy with being female. Out of the thousands of CAIS women worldwide, I'm aware of just a single recorded instance of one with a male gender identity (that one case on it's own doesn't mean a lot, as it could be due to chimerism, or a mutation to a gene somewhere that causes male brain development to take place even in the absence of androgen receptor activation).
By contrast, male babies who were born with a condition called cloacal extropy (which has traditionally involved reassignment to female as part of the surgical repair), had very high rates of dissatisfaction with a female gender identity (in fact it looked like an unmitigated disaster in the paper I read - about half of the patients who'd been reassigned to female had spontaneously reverted to living as male, and even the ones still living as female didn't sound happy about it).
What this shows is that your gender identity later in life depends on brain development that took place before you were born, and whether your identity ends up male or female depends on on whether or not there were androgenic hormones present during the time that brain development was taking place.
The thing I've been trying to make people aware of, is that doctors have for decades been in the habit of giving pregnant women drugs in doses that would suppress testosterone production in adult men, with most of the heaviest exposure to these substances tending to occur too late in the pregnancy to affect genital development or physical appearance, but during what appears to be the critical period when the brain undergoes its sexually dimorphic development. Either a male fetus is somehow magically immune to the effects of these drugs, or they've inadvertently created millions of people who look male but have female brains. Based on what I've seen of the effects of DES, it's the latter, which would explain why there's suddenly so many MTF trans folk about!
Whether medical hormones can also cause FTM transsexuality I don't know, however I suspect some can, and some of the replies to this thread appear to back up that suspicion.