What kind of company is this? I work in Korea and talking with other Koreans here there is a mandatory minimum by law. Asking for a day off is normal, but 5 days? That's extreme.
You are correct that the law states 15 days minimum with a cap somewhere like 25. But it's wildly violated and many South Koreans never get to exercise all of the days, or are artificially limited as to how large the blocks of time might be.
He works for a supplier of industrial equipment.