Especially non-European nations -- I'm under the impression that the names given to these countries, e.g. "the Maurya Dynasty", "the Qin Dynasty", "the Akkadian Empire", "the Old Kingdom of Egypt", etc. are modern inventions, not the names given by the countries themselves.
There seems to be a rather clear shift in naming once you enter the Middle Ages -- France is referred to as France, Travancore as Travancore, etc.
I know the answer for some of them, like the Greek city-states, for instance, or the empires created by them, like Macedonia. And then there's China, which generally believed in unification, so kingdoms were just considered candidates to rule the nation. But in places where fragmentation is the norm but so are territorial changes, like India, the Near East and the West, this isn't so clear.