The emperor in the story doesn't represent a specific person. In fact, it's an adaption of existing folk tales featuring foolish rulers transmitted via Spain, Arabia, Persia, etc. It's a very ancient theme.

But in the story, the emperor rode in a carriage and visited the theater. What sort of emperor was Andersen imagining? While the Holy Roman Empire had been dissolved several decades before the story was published, there was still an emperor in Austria. If I had to guess, Andersen was imagining one of those. Perhaps Francis the II and I, the Doppelkaiser.