Why 中華民國 = Republic of China?
The short answer: because Sun Yat-sen proclaimed it so. In a 1916 speech given in Shanghai, he asks the question:


  何以不曰「中華共和國」，而必曰「中華民國」？
  
  Why instead of 中華共和國, must one say 中華民國?


The reasons he gives boil down to drawing a distinction centred on the 民: saying that this is the fundamental difference between Qing dynasty feudalism and the Ancient Greek republic (and by extension, modern China). 

Although 共和 carries the idea of the res publica of Latin, he acknowledges this, and proceeds to give 民 greater importance.

As of early 2019, the only states to officially use 民國 are the Republic of China and the Republic of Korea.