It is the sensation created by the touch of the object which is suffering regardless of whether it is pleasant, unpleasant or neutral. The whole sphere of sensations is unsatisfactory.
External objects are not suffering until you come in contact and it creates a sensation. Anything physical object you have not experienced is not unsatisfactory to you. You don't know about it and not registered in your mental model of the world. For someone who experiences it, it will be unsatisfactory. To experience an object you need contact: the object, faculty and attention to the faculty.
Your mental model of the world is all that you have experienced. So everything you have experienced is unsatisfactory hence the world at large is unsatisfactory.
Also, the world is the 5 aggregates. Past experiences leave an impression in sanna and what is experienced now is registered through consciousness and recognised though sanna - which essentially builds your mental model of the world. Physical objects which are in your world is what is unsatisfactory to you. Objects in another's world unsatisfactory to them. External objects on their own are not unsatisfactory.
Essentially what is know wold to a being is unsatisfactory. Everything in the known world is unsatisfactory. Anything which is unknow becomes unsatisfactory then it is experienced, until then it is not known hence neither satisfactory nor unsatisfactory. What you crave for is also unsatisfactory as you might not get it.
The faculty (e.g. eye) itself is not suffering. It is the sensation produced which lead to suffering.
If there are an eye and an external object and attention which gives rise to eye consciousness. This results in a sensation due to contact.