In a way you can. Group the data you wish to copy. Ctrl x or c. Open the file into which you wish the group to be inserted. Ctrl V. Because it is grouped, it can be dropped, picked up again by another snap point and placed. This is a lot of circumlocution, for a simple process. I am assuming you are talking about the related technique used in Autocad, for instance. I use this quite often, as well as the Sketchup procedure I just described. Of course, there may be other variations.

In the scenes manager, make sure "visible layers" is checked, select the layers you want visible, then update, then save. If not, each scene changes with the current layer selection.