It's complicated. Even Taiwan doesn't recognize Taiwan (Republic of China) as a sovereign nation, separate and distinct from the Peoples Republic of China. There are Taiwanese political groups that do want that but they haven't gotten their way so far.
Either way, you probably meant that 13 nations have full and formal diplomatic relationships with Taiwan. Which, btw does not include the US, Japan or any EU nations.
It's not really that complicated.
After WW2 the Communists drove the Republic of China govt out of mainland China. So mainland China is controlled by the communists and Taiwan by the remnants of the Republic of China government. Republic of China government still claims to be the rightful government of all of China (Taiwan included), same with the Communist government.
And the people of Taiwan don't want to communist Chinese mainland govt to come and kill all their leaders, imprison people and take away their freedom, like the folks in officially happy Hong Kong. Adding on to your comment, it's pretty obvious why anyone would want to stay out of China's control.
Either way, you probably meant that 13 nations have full and formal diplomatic relationships with Taiwan. Which, btw does not include the US, Japan or any EU nations.