This question was spurred by a bit of browsing through Wikipedia, at which point the article regarding Trebizond caught my eye thanks to how it says "Roman Empire" on the right. Why is that? Furthermore, how could something as tiny as Trebizond call itself an empire, or was it a classification that was given to it later on?
Trebizond wasn't an empire by the conventional definition of ruling over people of multiple cultures separated by long distances.
Its entire area, depending on how you count, was only a few thousand square miles. Not that much bigger than New York's Long Island.
It called itself an empire because it considered itself (with some justification) to be a successor state to the Eastern Roman (lately called Byzantine, but that's not how they thought of themselves) empire. It was ruled by people from there and its (Greek-speaking) culture was similar. Trebizond hung on as long as it did because the area is highly defensible. But the Ottoman Turks, having taken Istanbul in 1453, decided to go to the trouble of taking Trebizond in 1461.