Internally people decide how to call themselves, whenever a government is established it has to determine what it’s governing. It will usually use a name that has some meaning to the region, for example when Spanish colonizers reached what’s now Argentina, they thought they had found a mythical river hat was supposed to take them to a mountain of silver, so they named the river “silver river” (Rio de la Plata), eventually when Argentina decided to become independent and needed a name for the newfounded country, they named the country by the river that was in it’s capital, and so silver river gave birth to silver land (Argentina, from latin Argentum meaning silver).

But more curious is how others name you, because you can pick your own country name and change it whenever, but other languages won’t. Take for example Germany, when the country was formed they named it for the historical name that means something like “land of the people” (Deutschland, from old German diutisc meaning of the people), but before Deutschland formed they were tribes, some people like the French and Spanish knew the Alemani tribe from that region, whereas others like Italy and England knew the Germanic tribe. So in Spanish and french the name of the country is Alemania, and in English and Italian it’s Germany. However Germany is in the geographic center of Europe, so almost every country interacted with that region before the country was named, and because other countries learned of that region through a country that had already met them, and those languages spread all over the world, making it so that Germany has so many names that there’s a Wikipedia page dedicated to it https://en.wikipedia.org/wiki/Names_of_Germany