According to the movie Mr. Miyagi was born in Okinawa, was interned in the US at the start of WWII, joined the US Army and fought in the European theater.

This is an extremely unlikely scenario, but not entirely impossible. Issei, first generation Japanese immigrants, were prohibited by law from becoming naturalized American citizens, only Nisei (born in the US to parents born in Japan) or later generations were elligible. As far as I can tell no Issei served in the US armed forces during WWII. While there were several regiments of Japanese-American soldiers during the war, they were made up entirely of 2nd generation or later individuals (nisei, sansei, etc.) That doesn't rule out the possibility of faking documents, though.

However, Mr. Miyagi's story has a few other kinks in it as well. Part of the reason why there was such a distinctive delineation of Japanse-American generations is because of major changes to US immigration law. The Immigration Act of 1924 effectively barred all immigration from Japan to America, and banned or substantially curtailed basically all immigration from Asia. The character of Mr. Miyagi was born in 1925, so he would not have been able to legally come to the US. Presumably if he did find his way into the US he would have done so under the guise of the other Nisei of his generational cohort.