When you wear “the white sweater“ is that the real you? Or is that a costume in the same sense as the black sweater? If so, where is “the real you”? Jean-Paul Sartre said that the essence of human Being, which he calls “Being-for-itself“ , is that it is not what it is, and is what it is not. He gives the example of a waiter, who must “play at“ being a waiter while he is on the job. But if he isn’t really a waiter,then who is?

Wearing either sweater is going to be inauthentic in some sense. Of course, your situation is a special case, because you have an unusual story which has made you especially aware of this. But feeling the imposter syndrome is an inevitable part of the human condition.