EDIT: I am obviously not being clear. I am not looking for a word for the common or familiar name. I am looking at how to describe the portion of the formal name that does not describe the system of government. In the name United States of America what is America? In the name French Republic, what is French? (Note that French is not a country name.) Perhaps what I am really asking for is the part of speech.
Most countries' names consist of adjectives, common nouns, and a word that refers to a people group.
Federal Republic of Germany, Czech Republic, Swiss/Helvetica Confederation, United Arab Emirates, People's Republic of China, etc.
How do you refer to the name part of the name?
I have considered toponym and place-name, but 1) this would still refer to the full name and b) even the full names aren't decidedly toponyms, because they refer as much to a people group and system as to a topographical feature. (Secondary question: can we consider country names toponyms?)
I also consider proper noun and proper adjective, but these would also refer to the whole country name, even though the whole name contains common nouns and adjectives.
Also, I am not looking for familiar name or the like. I seek a word for the part of the full country name. Say we are discussing the Czech Republic; I want to refer to the word Czech.