No. The government of Taiwan refers to itself as the ROC, because... it is.
Taiwan had been, since 1895, a Japanese territory. Following WWII, Japan was forced to give up most of its 'occupied' territory; Taiwan was included in this. While I'm not clear on the total sequence of events, it appears that the US initially took administrative control of the island from the Japanese, and ultimately transferred Taiwan to the authority of the Republic of China. It's not clear at this point if the ROC was administering the area in place of a local government (to be established later), or was being granted sovereignty over Taiwan. (The Qing Dynasty had previously controlled parts of Taiwan, but no nation exercised sovereignty over the entire island until the Japanese colonization drove into the mountains inhabited by the various aboriginal groups)
The ROC/KMT government took Taiwan as its own, though, and fled to Taiwan as a national redoubt at the end of the civil war, intending to eventually regroup and reclaim the mainland. That never happened, and eventually the military dictatorship of the KMT government liberalized and democratized. There was no "adoption" of the flag--the ROC has maintained the same Constitution, national symbols, name and so on. In fact, this is essentially mandated by the PRC, who has promised to respond to any changes with military force (as they would deny Taiwan belongs to the ROC--and thus to the PRC). This is also why you might have heard that Taiwan claims Mongolia as part of its territory, or says that it's capital is Nanjing--of course nobody in Taiwan actually believes that, but the ROC government is forced to keep those technicalities by the PRC.
To put it simply, saying "Taiwan calls itself the Republic of China" is a very misleading simplification. "The former government of China, the Republic of China, retreating from the mainland, saw its territory reduced primarily to the island of Taiwan, as well as other minor island groups (such as Kinmen, Matsu, and Penghu)."
tl;dr, small island isn't saying it's big country, big country's old government moved to small island, never stopped being that government.