My personal headcanon: The man who called himself "Khan" was in fact a junior officer in the Ubermensch crew, albeit a very quick-thinking one. When they defrosted him, he quickly realized they were expecting a *very senior* officer instead. Not wanting to be frozen again, he told them he was Khan*. The real, brown-skinned Khan is still in cryosleep.
*Notice he only called himself "Khan." Spock Prime was the one who called him by his full name. It is quite possible "Khan" didn't know the full name.