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In episode 2 of the anime, Head Maid says she claimed that the Demon King (under the guise of the Crimson Scholar) was a noble's child visiting from the Central Nations. However, in episode 9, Big Sister Maid (under the guise of the Crimson Scholar) claims that she was born a serf. How would the two stories be reconciled? Would they claim the first one to be another deception to hide the potential uncertainty that accepting a born serf's advice be prone to? Or would they tie the two together, saying that the Crimson Scholar was born a serf, but adopted into a noble family?
Anywhere in between. This setting is hardly an information age. Someone could probably move just thirty miles from where they were, claim an entirely different name and standing, and have no one be the wiser for years (as long as they have the wealth to back up their claims.)
There's also the question of what happens if they make known that the Crimson Scholar is actually a demon, the Demon King, no less. Wouldn't this invalidate Older Sister Maid's speech about being human in the public eye?
No particular reason that the overlap should ever be made public. Cook up any number of explanations for the resemblance, excuses for Demon King's image to not be circulated alongside the Scholar's, and/or allow Older Sister Maid to act as Scholar when/if ever they need to be seen together and the charade need never be broken.