I'm not sure about the actual legality of them asking him to stop, he's doing nothing wrong, he bought the game legitimately I assume, the person accountable is the teller/store/company that sold it to them.
That being said every year people get their Youtube Channels banned for uploading CoD ganeplay early. But thats a little different as YouTube works on behalf of Activision to stop it for copyright infringements which they could do for all videos, but they grow of the community so they only play the card whem stuff gets leaked.
But then again arguably every multiplayer match is different and the player is putting theor own skill into the game and thus it could be argued thst even they arent breaking copyright.
The gaming and copyright legal field is not very clear in my mind. Its not as different from the original as say uploading a cover to a song, but its npt as similar as uploading a whole song entirely copied.
Regardless im sure this guy - who looks like a moron by his profile pic and constant photos of his ciggarette close up - will be scared off by a legally written letter by Nintendo.