"People think they are buying a game (and rightly so)"
Most people DO buy a copy of the game as per normal sales laws. Such a sale has nothing to do with copyright and often the purchase is not from the copyright holder but from a store.
"and game producers think they are licensing you the rights to play the game (usually with lots of DRM to enforce these rights)"
Depending on country that is a nonsens sale since there is no need to purchase such a right, there is nothing forbidding you to play it to start with. Even if one do need such a license, and the sale is of a licnese, one still then needs to get hold of a copy of the game so that one can use the license. How would that be done? Well, as far as I can see, any license is typically sold together with a copy of the game. So one can in those cases argue that they sell BOTH a license and a copy of the game. In the end one to buy a copy of the game in either case.