If it's anything like kid icarus uprising, the "levels" are more akin to "Types". a level 5 power bomb has more uses/weight than a level 4 one. That way it's more efficient to spend 1 slot on a level 5 one than waste 2-3 slots on lower level powerbombs.
I thought those bars were weight bars of some kind, not exp bars.
Good call, totally forgot about KI:U's system. Weird to have a bar representing weight (you're totally right, it's weight - the same bar is on the bottom screen), when the number works just fine :/
I assume you'll just get different level ones by playing single player content, like you said, with higher usage/weight.