Why can countries officially recognise both the Democratic People’s Republic of Korea and the Republic of Korea as different political entities despite them both claiming to be the true Korea, but not the People’s Republic of China and the Republic of China, instead having to choose a side?
Partially because both of those agree on one thing and one thing only: There’s only one Korea. Recognizing one implies that they don’t recognize the other.
Having said that, some countries have managed, through some maneuvering and red tape, to have formal relations with both.