> both on the mainland and in Taiwan
This is an accident of history and a quirk of the English translation. Though the official language in English is described as "Standard Chinese," in Mandarin it's written as 國語, which just means "national language." The National Language described is Taiwanese Mandarin, 華語, which translates in English usually as Mandarin Language. By some measures it's very similar but it's different enough to deserve a different name, and the efforts of Taiwanese people to separate their culture from that of the PRC, and their ROC ancestors that came from the territory of the PRC, is a valid effort and should be acknowledged. Give it another 40 years or so and the languages will be quite distinct, as Taiwanese consciously incorporate more Hokkien, Hakka, and the various indigenous languages into their version of Mandarin to form a unique cultural identity.
> The Mainland is a well established concept when talking about China.
And people will happily, and inaccurately, describe the UK as "Britain" or "England," much to the annoyance of people who prefer to maintain their own unique cultural identity within the UK. In Taiwan this issue is even more critical as the sovereignty of the nation is under attack from many angles, including linguistically and culturally. There is no country on earth called The Mainland, and there is no country on earth called China, and CPC efforts to engage in cultural imperialism by casting a wide net around the concept of China and Chinese, as well as lay claims to historical Chinese imperial territories, and furthermore to imply that Taiwan is a PRC territory by calling the PRC the "mainland" (thus Taiwan as a province or colony), should be actively resisted.