Actor-Shared Background: Ralph Macchio is Italian-American just like Daniel LaRusso. In fact, the character was originally named "Daniel Webber" and was not intended to be Italian-American, but his name and ethnicity were changed after Macchio's casting.
There is serious implication throughout the series that Miyagi's style is Goju-Ryu (Hard-Soft Style) karate. First of all, his family name is "Miyagi", the same as that of Chojun Miyagi, founder of Goju-Ryu. There is a blink-and-you'll-miss-it moment when you see a photo of Chojun Miyagi on the wall among the pictures of Miyagi ancestors. A lot of the techniques Miyagi uses and teaches are lifted directly from Goju-Ryu (which, considering that Pat and Ralph trained Goju-Ryu to prepare for their respective parts, isn't strange).
In Cobra Kai: Daniel mentions that kata is the foundation of his karate. Goju-Ryu is based on two kata, Sanchin and Tensho, formalizing the "Hard" and "Soft" aspects of Goju-Ryu respectively.
Shown Their Work: Ralph Macchio has stated that Pat Morita took his role as Mr. Miyagi very seriously, and worked to develop the most authentic Okinawan accent possible to be respectful to the character.