Her medication doesn't change her biological makeup, science isn't that good yet. I'm happy to call anyone what they want, but surgery and drugs do not recreate your sex.
There's certainly lots of anecdotal evidence that many people's brains just seem to "work better" once on the correct hormones, and generally speaking transition has very good outcomes - people who have transitioned are generally doing better on key indicators, though still not as well as cis people. The people behind the studies believe that this might be down to how prejudice and discrimination affects trans people even after transition, though there's no further studies in that direction yet. (For a comparison, I believe gay people were doing worse than the general population on the same key indicators before general acceptance, and it's been gradually getting better.)
Here's a question, though: if we're not who we are in our own minds, who are we? Are you and I really only what others can see?
[0] A sex difference in the human brain and its relation to transsexuality http://faculty.bennington.edu/~sherman/sex/TRANSGENDER.pdf
[1] Male-to-Female Transsexuals Have Female Neuron Numbers in a Limbic Nucleus http://press.endocrine.org/doi/full/10.1210/jcem.85.5.6564
[2] Regional gray matter variation in male-to-female transsexualism https://www.ncbi.nlm.nih.gov/pubmed/19341803
In any case, it's actually more-or-less irrelevant in arguing whether a trans person is actually the gender they identify as, and is mostly a curiosity until it leads to more focused treatment. The bit that does matter is the last question - am I who I am in my head, or am I who you think I am?
Source?
If the difference is so clear in an MRI why not run a simple blinded study where researchers look at brain scans and tell you who is transgender and who is not? Such a study would not only prove this theory outright, but would likely make a researcher's career.
No. You can think yourself into a different gender. Gender isn't sex - this has been written about since 1963, and is generally agreed on at this point - gender is a combination of personal and cultural mental attributes, while sex is generally "what your body physically is". This messes up slightly when you get to certain sex-driven gender attributes, but, in general, it's a reasonably clear line.
Sex is mostly irrelevant unless you're intending to sleep with someone or you're a doctor, and even then what you're asking about is generally a certain set of characteristics where a number of people fall between the lines. If you're a trans woman who's undergone GRS and you say "my sex is male", that's going to confuse an awful lot of people.
Edit: My specific point, in this response, is that identifying as a gender which does not match your sex is not "playing pretend" or less important somehow, as you insinuated in your original post.
And personally, I see no reason why having male genitals is necessary for getting seen and treated as a man. If you behave like you want to get seen and treated as a man, I'll see and treat you like a man. The only people who have a remote interest in the actual biological makeup underlying that behavior are people who are romantically involved or interested. Oh, and medical personnel.
That's the long and the short of it.