Basically, it comes down to the fact that it takes Ubers to reliably defeat them without losing 1-2 Pokemon first. While there are a couple ways of doing it without Ubers (as others have mentioned) Wobby is basically just too good to leave off your team in a standard match if they are allowed. Much like Garchomp, Wobbuffet are Uber because if they weren't they would be on every team.

This is actually an improvement over GenIII battling where they were outright banned because if two Wobbuffet ever faced each other they could not hurt one-another or switch out. The game would go on until someone Struggled themselves into unconsciousness