Kumiko is making her first appearance in the The Karate Kid 'universe' since Karate Kid II (her only other appearance). At the beginning of The Karate Kid Part III, Daniel states that Kumiko was offered a dancing career in Tokyo that she could not refuse, much to Daniel and his mother's disappointment. Daniel and Kumiko had formed a romantic relationship in Part II.
Kumiko says that no other man had tried to fight to the death for her. This is a reference to the fight at the end of The Karate Kid Part II, which Daniel won after Kumiko was first taken hostage at a village celebration by the villain Chozen, who challenged Daniel to a fight to the death.
While Daniel and Kumiko are catching up over tea, she mentions seeing the Cranberries in concert in 1995 and lists a few songs the band performed, including "Dreams". This Cranberries song was used in The Next Karate Kid (1994), while Julie was practicing kata at the temple.
Sam leaves school after being punished, saying "This school sucks". Her father did exactly the same thing many years before, when he was scolded after a fight with Bobby.