What I have learned over the past decade is that there are three aspects which affect the way a person regards themselves: the physical aspect, the neurological aspect and the emotional aspect.
The physical aspect concerns such things as one's phenotype, being how one's body looks and functions. Here differences are plentiful, from the skeleton to muscle development and fat distribution. The skeleton itself shows marked differences pertaining especially to the pelvis, but also to the skull.
Also very important there is the reproductive system, which - starting with puberty - massively changes how one experiences one's body. This is also the area which mostly makes me realise that I am 'different', since I possess both male and female genitals, with my sex hormone levels being confused until finally settling on a female level. Whether one has a period, can become pregnant, or never has to concern oneself with such notions definitely changes a person.
The neurological aspect is intriguing, as well. Based on fMRI scans made over the past years, we can definitively say that there a 'masculine' and 'feminine' brain structure, based on the activity observed while performing tasks. While the correlation between phenotype and brain structure isn't very clear yet, it's however obvious that there are two main development types of the brain. One displaying a tendency to reason and reflect before taking action while also possessing strong multi-tasking capabilities, and the other being more direct in how it processes input and decides to perform a certain action.
Finally, that leaves the emotional aspect, or the part where one can definitely say that one 'feels like a woman'. This part is an amalgamation of the former parts, along with one's experiences in life. As a result it is as unique as the individual themselves. Someone raised in a certain environment will adopt different habits and assumptions - thus changing the way they think and feel about themselves - than someone raised in a very different environment.
In the end this means that 'feeling like a woman' is a very meaningless phrase. Yes, there are certain physical and neurological characteristics which would cause society to stuff one into one of either binary box, or provoke disparaging remarks if both 'do not match up' (e.g. tomboys and 'girly' boys).
Personally, the physical part has made it more than obvious to me over the past decades that physical characteristics including reproductive systems exist on a spectrum. Neurologically I seem to be in possession of the former type of brain, though looking back I cannot really say that I couldn't have been a guy with this brain if I had been born with a regular male body. Frankly, I much prefer guys with a more strongly developed 'feminine side', as it's usually referred to.
The definition of a 'woman' thus is rather hard to pin down. No clear-cut definition is found in the physical aspect, or the neurological aspect. Further, it's patently ridiculous to ascribe a binary notion to the emotional world of an individual.
In the end the only conclusion one can draw is that we're all individuals, with individual bodies, individual minds and our own experiences and thoughts. The so-called 'gender divide' is more of a self-fulfilling prophecy than anything grounded in biological or scientific fact.