Erasing the blackboard is simply a set phrase which is understood to mean erasing everything which is written or drawn on it. This is a frequent activity. When we talk about paper, it is extremely rare that we erase everything on it, and even more rare with an entire notebook. We therefore do not need a shorthand way of saying this - we talk about erasing particular items on the paper or in the notebook, even when we want to erase everything.
To erase the blackboard would be a shorthand way as you pointed out while to erase the notebook cannot because we never want to erase all the things in the notebook as you also suggested. which really makes sense to me.