No.. technically it's not a C&R.
The handy thing about having a C&R, is the benefits one who is buying C&R eligible arms across state lines and avoiding FFL transfer fees.. It ships to your home and you log it in your own log book. I've had one for a number of years.
As for having to have a C&R to own a C&R weapon.. no, many people own Garands, Luger's, Mausers and a multitude of others with no license. It's like any other weapon then.