Not really. Asian nations obviously also had hats but they werent as significant as a crown.

Chinese empires would punish people for wearing golden robes as it indicates a rebel trying to usurp the throne. Nobody really talks about the hat.

There are other symbols similarly to the globe apple of Holy Roman Empire of Germans and England.

Mongols were heavily influenced by chinese culture even though they have significant differences. Mongols did adopt foreign customs though so Mongols in the west did adopt crowns to signal authority over the new subjects. But in Yuan Dynasty and in Mongolia the hat wasnt as important as a crown.