I have looked through the list, and yes, they all do, with the exception of Magikarp, since Splash is a Normal Type move.
Why? No non-GameFreak/Pokemon Company member could say for sure, but I would guess it is to prevent confusion as to what types your moves are. This way, you know that, say, if you catch a Gardevoir, the move's Type is Psychic. It helps sort them.
Pokemon Ranger has a similar thing, where Pokemon are sorted into Groups instead of Types, and their PokeAssist is the same type at them.