There are two parts of this question. The first is why the term "Republic" is involved in the title and the second is why the word "South" is not in the title.
"Republic" is part of the name since South Korea has a republican form of government, with a legislature and presidency. During the Cold War, it was an authoritarian republic, but since 1987, it has been a true democracy.
"South" is a laymen's way of distinguishing the two governments in Korea, because the democratic government controls the territory to the south of the communist government. However, the South Korean government claims that it is the sole legitimate government in Korea and reserves the right to control the entire peninsula, not just the southern half. (The North Korean government has the same view, that it reserves the right to control the whole peninsula.) As a result, the official name of the country reflects that the current government's ideal is to rule the entire peninsula.
Related Q&A：
Why is South Korea called the Republic of Korea？-Well, you know, South Korea is officially known as the Republic of Korea because of its political system and form of government. The term \Republic\ indicates that it's a country where the power is held by the people and the government operates based on certain democratic principles and structures. It's a way to distinguish and define the nature and governance of the country. Oh, and this name also reflects its status and identity in the international community. So, that's basically why it's called the Republic of Korea!
