It's not surprising that it peeled off, though it is surprising that it lasted as long as two months: based on your description, that is the kind of film that is often applied to products (particularly over glossy surfaces) during or after manufacture. As often found on printers and music players. It isn't supposed to be left in place!
If the outer surface gets marked, look after it more carefully and/or buy a more durable notebook. None of mine have any noticeable marks and one of them is two and a half years old.
If you're really worried about the screen, apply a proper screen protector, cut to size, which you can get from various suppliers around the web.