It should be “playing with Nintendo” instead of “playing Nintendo” toe be fair. To be honest, “playing with Nintendo cards” would be the most accurate, but “with Nintendo” is still accurate enough and still gives the sentence the desired effect. But no, “playing Nintendo” isn’t correct. Unless they made some specific game variant and included the rules with their cards or something.

He would be playing hanafuda, which is the Japanese card game that Nintendo was producing at that time. Not as funny as imagining Dracula trying to beat Ninja Gaiden or whatever.