Characters cannot act differently than they are written and it seems pretty clear that the errors are from the production, not in-story, though even in-story the notion of Khan being Sikh is very tenuous at best. It’s essentially a line of dialogue and a painting that doesn’t actually depict a typical Sikh man.
Come to think of it, the line in Into Darkness where Khan says Harrison is a fiction created by Marcus offers almost as much weight of evidence to infer some cosmetic surgery was done as the evidence in Space Seed suggests Khan is a Sikh. Both are conjecture, neither one more compelling than the other taken in isolation. Constant rewatching of TOS has more to do with accepting McGiver’s speculation as fact than anything actually presented.
Come to think of it, the line in Into Darkness where Khan says Harrison is a fiction created by Marcus offers almost as much weight of evidence to infer some cosmetic surgery was done as the evidence in Space Seed suggests Khan is a Sikh. Both are conjecture, neither one more compelling than the other taken in isolation. Constant rewatching of TOS has more to do with accepting McGiver’s speculation as fact than anything actually presented.

