No, you only need to add it once (preferrably to game.script).
In WME, each object (game, actors, entities...) can have their own items. That's why you can use both:
Game.TakeItem("something");
and
actor.TakeItem("something");
The Game.InventoryObject property then specifies, who's items are displayed on screen.
It's intended for games with several playable characters, to allow you switching between them.