At the time of the Korean War, Korea was one single country.

The only seperation was an administrative one, after WW2 Japan gave up Korea, and two Allied countries were given the task of overseeing the transition from Japanese rule, to independence. The US in the South, and their (at the time) ally the USSR in the north. It was just a made up line.

Then the US ends up basically installing a US-friendly brutal dictator into power, and claimed it to be the government for all Korea.