You can use the menu designer to override the default action for Edit->Copy (which is called from the shortcut key). The default action will copy the full screen if there is no selection. Always been that way, so would probably break something or upset users to change the behavior. Here’s a script you can use instead of the default action. Just place this script, including the [ and ], in the action field for Edit->Copy:
[if activesession.getselection(0,0,0,0) then activesession.copy]
You also need to place the same script in the Toolbar Copy button and the context menu Copy item. Else those will have the original default behavior.