I am Taiwanese. I was born in New Zealand but was brought back to Taiwan to be raised there up until the start of High school.
In my honest opinion, China claiming Taiwan is not a country is absurd. But to get to that conclusion, we'd need to look at history.
The Republic of china (A.k.a. ROC) Had come to power by overthrowing the imperial rule of Qing Dynasty and replaced it with a Democratic government (Only the so called 'Blue' party was allowed until 1980s). At this point, Taiwan is still unified to China's Mainland. It is then invaded by Japan (WWII) and the government, in a treaty, signed over the ownership of Taiwan to the Japanese. After the war, the Japanese surrendered, their troops and Governing officials left and the power was given back to the Democratic government. During the war, In China, they were only able to hold back the Japanese advances by the combined power of the army of ROC and The 'People's liberation Army' which is Communist. The communists and the Democratic Capitalist government disagreed with each other so much they waged war on each other. It is unclear which side had started the civil war. At this point in history, people living in Mainland china started choosing sides, unknowing the complete disconnection (many families ended up split on two sides of the strait, unable to write letters or travel to the other side of the strait) the end of the civil war ended with. The civil war left ROC with control over Taiwan (This may not be argued to be false if any logic is applied) And Mainland China is now under control of what is now People's republic of China (PRC). Up until 1971, ROC (Taiwan) held a UN membership as China. This is not 'stealing' the power from PRC since the PRC was completely disconnected from the rest of the world up until than and after all, the ROC had been the official government of the entirety of China until a civil war torn it apart.
Now, why does the most part of UN not recognize ROC as a country? The answer is simple. The PRC, upon replacing ROC in the UN seat marked as 'China' carried on the hatred of the ROC and every nation knows, the PRC will never allow ROC in the UN and Mainland China, being a world superpower has overwhelming influence over many countries. Most countries did not bother arguing against China, Including the US.
As you might have noticed, the topic over wether or not Taiwan should be a part of 'A Greater China' is very controversial. But as a final effort to persuade you that Taiwan should be a State on it's own right, if you own one big piece and a small piece of a candy, which was from the same package. If another person comes along and buys the big piece, does it automatically gives the person ownership over the small candy?
Living in Taiwan and Taiwanese history education (basically Social Studies)