As others already mentioned, Uber does work in SKorea (or at least Seoul), altough it's not really an uber, afaik its just a proxy for kakaotaxi while using Uber's interface
Korea uses KakaoT as a ride hailing app. But all it does is hail taxis. Uber in Korea just hails Uber branded taxis. I have no idea if they are officially affiliated with Uber or not.
Uber operates through a local JV with Tmap called UT. Taxi drivers typically sign up for both Kakao T and UT, except when exclusively branded. (Kakao T and Uber both operate branded taxis.)