If you collect the item then that one instance of the object is set to 1 and you store it as 1, but the other instances presumably remain as 0. Then when you load the game and get that item from the local storage you now set all instances to 1. Not understanding that last event. Even if this did work are you really going to repeat those events for each collectable in the game? Also I think you should not be saving instance variables to local storage, probably why you found no information on this.
If you're doing collectables I would advise using an array where you can list all collectables in Y=0 and then have yes/no or 0/1 in Y=1. Then you can save the full array as JSON format into local storage then you only have to save/load one item. On starting the game you load the local storage item into the array and it's filled with what you have collected, 1s and 0s.
Or if arrays sound like it's not your thing then please revert back to global variables, that's all you need to store to say that you collected an item. When you overlap the object, you set a GV to 1, not on the object itself.