Let's just put this in the "for what it's worth" category as long as the debate about the "proper" person to play Khan is going on.
Most Sikh males have Singh as their last name. So in "Space Seed", Noonien Singh would be his first and last name, and Khan would be his title. But I can't find any use of that title in Sikhism. The title came from the Mongols and was adopted as both a title and a surname among Muslims in central Asia (including India). It is not a Sikh or Hindu title or name.
So, the whole thing may be screwed up six ways to Tuesday in every aspect. We have a character who was a Sikh tyrant taking a Muslim title being played by a Mexican in TOS and by an Englishman in STID.
While you have the information here based in fact and history, the basics are that this was a fictional character from a television series.
Beyond that, in the fictional sense, the world in which Khan ruled could have had so many messed up customs and cultures by that point, even being that Khan was "ruler than more than a quarter" of Earth, he could have taken whatever titles he wanted - who would have disputed him?
Oh to be sure, there are thousands of ways to rationalize the title in the Trekverse. It's not that big a deal.
However, out of universe, it does make me wonder about the choice of the title during the writing of "Space Seed". Was it a deliberate choice to mix them to show that chaos probably existed at the time, or that Singh chose it for political reasons, or was simply an accidental or careless mixing up of two different religions and cultures by the writers that made it all the way to shooting?