Republican and imperialism(historical meaning, rule by an emperor), were the two options available to the revolutionaries when KMT founded Republic of China.
Note that republican government is a very broad idea and various groups of people has their own version of understanding. But back at that time those who win over the other side has one in common, that they oppose the ruling by the emperor.
民国 and 帝国 are a pair of notions more understood by Chinese in 20th century when the ROC was founded. That's one reason 民国 is used instead of 共和国. Both Republic of China and Republic of Korea chooses their official translation to be 民国 to differentiate themself from 帝国 such as Qing Empire, Japan Empire or Korean Empire.
Also back to 19-20 century, many east Asian theorists and politicians borrowed the idea of nation 国族 and nation state 民族国家 from their European contemporaries.
On the other side, some people instead support pan-asianism which promotes the unity of all asian people.
Both of them aim to save asian countries from European colonists and preserve independence.
KMT was a nationalist party who want to construct a new national identity and a new nation state. And the south Koreans oppose the pan-asianism as the latter is believed to lead to the invasion by Japan Empire.
For this reason, they want to emphasize that the new countries, ROC and ROK respectively, are nation state owned by their own people.
To sum up, 民国 emphasized two things:
Republican government - as oppose to a government ruled by the emperor
Nation state - as oppose to a country manipulated by foreign nations