For me it's the way the games have roots in reality. The story takes place in real settings that start at a real time. It gives an amazing sense of freedom but doesn't go outside of the rules of the story. Unlike what other games have done, you wouldn't have Ryo pull someone from their car, punch them in the face, take their money then run them over with their own ride as they pick themselves up. Ryo wouldn't do it in his story and neither would you in real life.
That's why it connects and resonates with us because as we play, it puts us in Ryo's shoes and makes his story our story. The whole experience starts to feel like something real which allows you to invest yourself completely.
Take all that and wrap it in all the insanely high production value that's already been touched on here and what you have is a gaming experience that simply stands alone. it's more than a game. It's Shenmue.