The United States is the same country today as it was in 1776 because (most obviously but among other things) there has been an unbroken continuity of government. In some places like China for example they look to language or culture for continuity but the US by its nature is a more diverse and amorphous place. The United States Constitution is really what makes America America.
On the other hand, claims of links to ancient kings and kingdoms has long been a trick of governments and nobles to try and give themselves some kind of legitimacy. Of course, 99% of the time these claims are totally bogus and made up.
This also brings to mind Palestine. Many people believe that there is some connection between "Palestinians" and the very ancient Philistines. In fact, people in this region for the most part have about 0 genetic or cultural link to the ancient Canaanites. It's purely a politically expedient assertion to claim otherwise. The idea that there was a region by a certain name here has persisted but is that really a basis for a country? When the Romans ruled over the Levant they named the Jewish-dominated area around Israel as Palestina, a cognate of Philistines, because the Philistines were the ancient enemies of the Jews according to the old stories. It was meant to undermine Rome's political opponents in the area, not as a reflection of any real connection or continuity. and now...
By the way, you're right that in Mesopotamia there was a lot of movement and change, but the people and identity didn't all get wiped out like some Native American groups, just mixed around, shifted, and changed gradually over those 4000 years. Iraq isn't Akkad 2.0, but the lineage is there... so maybe Akkad 6 or Vista :)
You should have put both Koreas in one, since the north and the south cam BOTH from Gojoseon and Jin.
Also, a majority of Gojoseon's lands are at modern day PR China. Why not put china too?
except that, i enjoyed the quiz.