As a youth who has participated in both I can say that, though both produce some similar sensations, they are definitely not one and the same. The fainting game produced (in me and all of my friends who participated) about 1-5 seconds of unconsciousness, with every friend claiming that this was the "effect" of the "game"- it was basically the so-called fun of the game... passing out for seconds and going into a semi-lucid dream state that felt like an extended reawakening of reality upon coming back into consciousness (as if the moments preceding the faint were part of the dream experience.)

Inhaling nitrous oxide did also put you in a sort-of dream state but the "high" was more of a sensory-motor change than a loss of consciousness.