Part of, not the whole
I am who I am, in part, because of my quirks. Saying they are not part of me would be denying who I am.
I don’t understand what’s offensive about people being happy with who they are.
Question: isn’t identity that which makes one distinct from others, like name or government identity number?
Not at all. You're judging by the standards of neurotypicals.
Suppose we discover a genotype that correlates with extremely high social competence, people who are just naturally much better at communicating and working with others. (This isn't much of a stretch; we know there are people like that; we just don't know if it's genetic.) If we document this new, higher level of social functioning, does that mean that neurotypicals are now also defective, diseased?
For this to be true, autists would have to communicate and/or work with other autists as competently as neurotypicals mesh with each other. Strictly speaking, we should be able to work with other autists more easily than we work with neurotypicals who have a basic familiarity with autism, since we also have that advantage. In my experience we are not. Replacing your left hand with a sword also has measurable advantages but that doesn't make it not an overall impairment. Personally I find this cultural trend where "your deviation is your identity" to be illogical and offensive.