According to google translate and other dictionaries, "republic" means "共和國" in Chinese.
The short answer: because Sun Yat-sen proclaimed it so. In a 1916 speech given in Shanghai, he asks the question:
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.
民国 and 共和国 mean exactly the same. However, nowadays we prefer 共和国 to 民国 as the translation of “republic”.
Etymologies (word origins)
The native Chinese term 民国 is actually a mimicked term from 王国 and 帝国.
王国 “king’s state”, i.e. kingdom
帝国 “emperor’s state”, i.e. empire
民国 “people’s state”, i.e. republic
The Japano-Chinese term 共和国 is a Chinese term borrowed from the Japanese term 共和国 (きょうわこく), which means “gonghe state”. The Gonghe Regency (西周共和时期) was an interregnum period in Chinese history from 841 BC to 828 BC, and China had some similarities to a republican state during that period. As a result, “gonghe state” is modernly used to refer to a republican state.
To sum up, 民国 and 共和国 mean exactly the same.