The answer is no, Taiwan didn't claim anything. Taiwan is Taiwan. Its the Republic of China who still claims China, Mongolia in its territory in the constitution, but lives in Taiwan illegally.... Does it matter to China that Taiwan doesn't want to be part of China?
Both the ROC and the PRC still officially (constitutionally) claim mainland China and the Taiwan Area as part of their respective territories. In reality, the PRC rules only Mainland China and has no control of but claims Taiwan as part of its territory under its "One China Principle".
On August 24 and 25, 1958 Chinese Communist and Chinese Nationalist forces clashed in the vicinity of Dongding Island, which the Nationalist troops controlled. The action was seen as an attempt by the communists to land on the island. This was the only naval and amphibious landing action during the crisis.
The other main island chain, Penghu, is an archipelago of 90 islands and islets. Lying off the coast of Taiwan, these would be a deadly obstacle to any invading force. Extremely well fortified, the largest islands bristle with anti-ship and anti-aircraft missiles, early warning radar systems and well-trained troops.
Taiwan is a highly advanced economy producing an estimated $786 billion in goods and services in 2021. The United States and Taiwan have deep and growing commercial, financial, and trade ties, which advance U.S. interests and help create economic opportunities in the United States.
As a leading democracy and a technological powerhouse, Taiwan is a key U.S. partner in the Indo-Pacific. Though the United States does not have diplomatic relations with Taiwan, we have a robust unofficial relationship.
In reality, the PRC rules only Mainland China and has no control of but claims Taiwan as part of its territory under its "One China Principle". The ROC, which only rules the Taiwan Area (composed of Taiwan and its nearby minor islands), became known as "Taiwan" after its largest island, (an instance of pars pro toto).
Current status Today, Taiwan is a democracy. China is ruled by a communist government. The idea of freedom in communism is different than how people in a democracy think freedom should be. Few people in Taiwan want to become part of Communist China.