To some extent I agree and to lot I disagree. Yes, mental illnesses are misdiagnosed and this trend of peers diagnosing themselves or people around them is way more alarming but there are people who actually have those serious issues, especially when you battling for your life, as you said, they give shape to something you cannot make sense of without the diagnosis.
I, myself, after years of battle with chronic mental illness is trying to leave those dark chapters behind. But separating ones identity after long treatment and struggles is totally different subject from misdiagnosis.
And no good doctors ever say that people with chronic diseases are incapable of change. They say, there is no guaranteed cure but we all can learn to manage our symptoms and hence, get back to life on our feet like any other person.
Diagnosis is important and it is highly likely for a while it will become their identity because it gives them a shape to a chaos but leaving it behind is also important but on their own speed, not because somebody wants them to or demands from them. Healing is a journey, not a destination!