Miyagi-Do seems to be a loose interpretation of the karate style Goju-Ryu. Most karate styles tend to have several different katas, called “forms” in other martial arts, but for some reason, Miyagi-Do only has the one. The kata is taught to Daniel Larusso by Mr. Miyagi in The Karate Kid Part III, and LaRusso goes on to teach the kata to his students in the sequel series Cobra Kai. That kata includes signature moves seen in the Goju-Ryu kata Seiunchin, though the kata style is seriously toned down for the film.
Chozen, played by Yuji Okumoto, in The Karate Kid Part II is Mr. Miyagi’s nephew and one of the villains of the storyline. The whole family trains in the same root style of Miyagi-Do, and the patch seen on the front of Chozen’s Gi in the storm scene is a signature Goju-Ryu patch. Furthermore, the history of Miyagi-Do, as Miyagi tells Daniel while in Okinawa during Part II, is similar to that of Goju-Ryu’s origin. Miyagi explains that the founder of the martial art passed out drunk on a fishing boat, woke up in China, and trained in Chinese martial arts for ten years before returning to Japan. The founder of Goju-Ryu, Kanryo Higaonna, according to The History of Fighting, trained in Chinese kung fu while working in China and brought it back home to blend it with karate, creating Goju-Ryu. Miyagi’s drunk fishing boat story, as comical as it is, didn’t happen to Higaonna.