I don't think you even need to get into the "race is an illusion" argument to self-identify as a different race. I remember when I applied, all the applications asked the question with phrasing like "Which race do you consider yourself?" I wondered if there would be any consequence if I answered Black or Hispanic (I'm white). Probably not, I'd think, as they can't prove I don't "consider myself" one of those.

Surprising that it doesn't happen more often. Or maybe it does...