Sort of, except not really, except yes really. It's complicated.
The China that was a founding member of the United Nations was the Republic of China (ROC), and it controlled both mainland China and what we call Taiwan. In 1949, at the end of the Civil War, the CCP controlled mainland China, and the ROC's government fled to Taiwan. Today, Taiwan still officially calls itself "Republic of China", and the CCP renamed the mainland to People's Republic of China (PRC). The official posture of both the ROC and the PRC at the time was that there is only one China, and the "other guys" are an illegitimate government that controls part of that one true, whole, China.
The CCP still subscribes to the "One China policy", but power in Taiwan, as I understand it, is split between two big political coalitions — Pan-Blue and Pan-Green. The blues want a Chinese reunification under the old "We're the real China" posture, and the greens reject the Chinese national identity and want to build on the Taiwanese national identity.
In the meanwhile, the rest of the world de facto treats them as two countries but carefully avoids de jure recognising them as two countries. Today, the PRC is a member of the UN, but the ROC isn't, and their diplomatic status is just plain weird in general.
