The Unknown does not give his real name, but two fake options: Erostratus and Empedocles. What do these names reveal about his character?

He is willing to die for fame

Explanation:

Based on the given statement, it means that the character wants fame by all means even if he would die at the end of the day. He is looking for fame and he is also ready to die for fame. He doesn't let people know his actual name so he used fake names so that no one would have a clue of his real name and where he is from.