Is "Emperor of Mexico" the same as "Mexican Emperor"
Solution 1:
Without any other context, "Mexican Emperor" would normally be interpreted as "Emperor of Mexico". But it can be used in other contexts, like "Mexican Emperor of America", which would then mean the Emperor of America who is Mexican. Or in answer to the question "Which Emperor of China are you referring to?", the answer could be "The Mexican Emperor".
This is a common ambiguity when using a nationality as an adjective modifying a role. It can be used to qualify the role itself, or to qualify a person fitting the role.