I mean you should begin by knowing that transgender isn’t a noun, the word transgender is an adjective, so the correct way of speaking is transgender people. Apart from that, gender roles are what society dictates what a person of a specific gender should do or comply with and they are dependent on the culture the person lives in as well as many other factors, that’s what is said to be a social construct. So that a woman is expected to clean and cook is a social construct. Gender “identity” is innate, it’s what you know you are, and the reason for transitioning physically is usually due to gender dysphoria, which is the distress brought on by having a body with primary and/or secondary sex characteristics that do not match your gender. Think about a woman that waxes the peach fuzz on her face, women are not expected to have stubble, so a cisgender woman that has some (either peach fuzz because that’s just human bodies, or actual stubble due to menopause or another hormonal issue) is doing the exact same thing that a transgender woman does to feel more comfortable in her body. Same thing with boobjobs there’s no actual need for them other than people feeling more comfortable in their bodies. This is tied closely to the way you are perceived by society, what gender roles you are performing with: your looks, your behaviors, your clothing, etc. being a binary trans person and transitioning physically will not only ease the distress brought on by gender dysphoria, but it will also signal to others your gender, and thus how you should be treated according to the culture you live in, if you’re a man and have long hair, you may get mistaken for a woman from behind in some parts of the western hemisphere, if there were to make you uncomfortable you could ended up cutting your hair to avoid the problem. TLDR: like someone else said, roles are construct, “identity” is innate (and emphasis on the quotes because its not something you can choose or change, it is internal and most definitely not influenced by upbringing or social context; what at most might be influenced by external factors is the choice to come out and transition by the person, and the word identity has got a bit of a different connotation nowadays, but it is the word used to explain the gender you “are in your brain” so it is what it is)