Is it generally allow for 5 person in one taxi when travelling around various cities in Korea?
No. The majority of taxis are Hyundai Sonatas. Driver, front seat passengers and three in the backseat all that will fit properly with seat belts. We’re you
Planning on having someone sit on another passengers lap? Vans around normal streets are not as common. You may want to find a taxi app. My sister uses a taxi app and perhaps they can send a van instead.
As 4k said - 100% impossible so you will need to use a Call Van taxi every time you want one. But if you're going short distances, taking 2 taxis isn't that difficult or expensive.
Instead - if you want to stay together - consider hiring a private driver for a day make it easier.