Not all Items can be equipped to all Characters, mostly because this item won't take any effect on that specific friend or because it would make the game unbalanced.
The same goes for game modes. While you're playing Solo, for exemple, Defibrillator item won't work. That's why you'll see a red warning every time you try to play with it on a battle.
I'm sure with the best set of items, you're going to be the king of the Zoo!