counter for books, volume
册 centers on bound collections of pages: a volume or a counter for books. It represents a physical book-like object made of stacked sheets.
册 is a pictograph of bound bamboo or wooden slips used for writing in ancient times. The vertical strokes represent individual slips, and the horizontal strokes represent the binding cords. The modern form is a simplified version of 冊.
The vertical lines are pages, and the horizontal lines are the binding. Picture a stack of paper tied together into a book: 册 is a volume.
For サツ, imagine a stack of books sitting on a shelf: stack -> サツ, and each 册 is one volume in the stack.