A study of DNA can reveal a person's lineage. A study of teeth can reveal where they grew up. A search of the word Sikh on Google generates images of men and women who look nothing like the actor playing Khan. Like it or not, there are physiological differences in the appearance of people that identifies where they are from.
I believe to be true that we don't know for certain that Harrison is Khan because there is no canonical footage of the discovery, retrieval, and manipulation of Khan and his people by the Federation. My options are, do I accept what I am told by Harrison or do I not accept what I am told by Harrison? Is there evidence from the films to support either option? Are we shown Kirk's crew attempting to verify this man's claim? Like I said above, DNA can be used to trace lineage.