It’s feasible, but we would need to be careful to make it clear that this is a platform feature so that apps don’t then implement their own notification–or else you’d see two! I also think this shouldn’t be implemented using standard system notifications as those are rather large and it would get really annoying to see big chunky system notifications appearing and covering up content in your windows all the time in response to a very common action.
Maybe the Clipboard icon in the system tray could do a little shake or other animation or something. Like a little animation of something getting sucked into the clipboard when cutting or copying, and something flying out of it when pasting, maybe.