Is it possible to represent the same object differently in different windows?
Yes, easily. When an object is first loaded, it will be displayed with the defaults for that object type, and these are the same for each window. However, when you choose to format an overlay the command is specific to the selected overlay in the active sheet of the active window so any changes will only apply in that view (unless the “Apply to all overlays displaying [object name]” box is checked).