How I think this will be implemented
I was thinking of the possibility of this ever since the sticker power up update, where stickers boosted the power of fire based attacks, resistance to slashes, etc. I thought they might implement this for PT's Pokemon, and maybe the other Pokes in the game.
I think only the Pokemon have bodily weaknesses and resistances, but others' attacks will be strong or weak against the Pokemon depending on the attack's type. This may mean that Charizard is lighter than we think as well, because he got hit with a fire attack, which he should resist. However, he may be neutral to his own type in Smash.
The idea that random non-Pokemon characters would be assigned a Pokemon type is far-fetched at best. I don't think we have to worry about this being a balancing issue.
The Pikmin stuff (electric resistant yellow Pikmin) makes me think that this elemental system may extend to Pikachu, as well as Lucario and possibly Jigglypuff. Pikachu may have an innate advantage over Squirtle and Charizard if this is the case. I don't know how it would apply to Lucario and Jiggs, since they are weak to Fighting type attacks, which may be a LOT of moves.