I can't get over the fact that apparently no one has ever heard the phrase "this isn't your father's <XYZ>"!
When he says, "This isn't your father's Mario Kart" he means it isn't the same as the previous games and should at least have some noticeable differences (usually improvements but specifically differences). For example, you could easily describe Mario 64 as not being your father's super mario.
It doesn't have anything to do with not playing with your dad, trust me. DadPlay, or "thrashing the old man who can't really play games but likes to show some interest" will still be available in Mario Kart for Wii!