It would work with a backpack, or traditional schoolchild's satchel worn on the back, or even a bag that's worn on one hip with the strap across your chest . Anything type of bag that you wear on your body - as opposed to carrying in your hand - can be put on and taken off. If you hold in a bag in your hand, then you 'pick it up' to take it somewhere, and then 'put it down' when you've arrived. In other words, if you wear the object on your body, you put it on and take it off. If you carry the object in your hand, you pick it up and put it down. I hope that makes sense.

In English, we do say "put on a piece of clothing" and "take off a piece of clothing". In the United States, we use backpack (if you're referring to the bag you use to carry your books and things back and forth to school) more than bag (although you may hear it used that way from time to time).You can say "take off a backpack/take a backpack off" and "put on a backpack/put a backpack on". I hope I explained it clearly. Let me know if I didn't and I'll try again :)

No, you can carry or wheel a bag but you cannot wear one. You can put on a backpack and take it off.

Yeah, backpack makes more sense. It would be better to say 'put on the backpack' and 'take off the backpack'. If they own the back pack you would say 'Put on your backpack' and 'Take off your backpack' I hope this helps :)