Say "marry" - like "I want to marry that person" - and add "o" at the end. That's Marry-o, Not Mary-o
Mary is pronounced different to Marry.
Say "Maw" like "Saw", and add "rio" - like the City in Brazil. That's Maw-rio.
Devils advocate - I work with an Italian named Mario and he doesn't pronounce it Maw-rio. He says "Marry-o"
I've only ever heard North Americans pronounce it like that.
On this side of the Atlantic I've always heard it as Marry-o. Which is how Charles Martinet says it in the games.
The difference is the pronounciation of the letter A sound in the word.
The way RMC said Mario first in the video is how I say it and how I hear Charles Martinet say it, and how a genuine Italian says it. When he "corrected" his pronunciation to "Maw-rio" that to me sounds wrong.