Well apart from over 70 years of mostly stable government (with Tiananmen square being the exception) they are broadly recognized by most countries and the UN and have been for close to 50 years.

There's little evidence of support in China or even in Taiwan for the idea that the Taiwan government are the legitimate government of mainland China, and Taiwanese ambition is pretty much limited in practical terms to self government.

Indeed, I don't believe Taiwan claims China in anyway I can find reference to.