Err, no. You only own a license. You never own a game/movie/song. Unless you pay millions of course.
You probably are confusing it with "renting vs. sale". Digitally bought licenses are actually better than those tied to physical games, because you can never loose them. (Not that it helps you much).
For the purposes of law, courts apply the exhaustion doctrine to "shrinkwrap" licenses, so any packaged goods containing software are deemed to be an exhaustible good and therefore "owned".
There is the right of exhaustion on the one hand and the right of reproduction on the other - we are indeed forbidden from reproducing the right of use, but then applies to most goods. I could buy a Rolex watch and sell it on, but I'm forbidden from making a replica of it. The difference between a Rolex watch and a piece of software is the technical hurdles in replicating it and distributing it. The (lack of) right to do so is effectively the same.
I grant that digital ownership is another thing altogether, but the generally held position (including that of the ECJ) is that the exhaustion doctrine cannot be applied to downloaded software not because there is no physical product but because the enforcement of the exhaustion doctrine when so many reproduced copies are made is nigh-on impossible. If A sells a license plus software per download to B, then B acquires a copy of the software but A still retains that software on their server. Hence the problem of reproduction.
I'm aware that this basically argues against the OP's point - by granting a user a digital licence on the basis of a shrinkwrap licence, GOG would effectively be granting a second fee-bearing licence at no charge and would owe the copyright owner money for this.
You still don't own the game. You still own the license. Vastly different.
Edit: Maybe I should elaborate. You own the disc, the box in retail. And you own the license which is directly derived from the physical copy.
You never ever own a game. You never own the source of the software or music your license is derived from. GOG also doesn't own any games, they themselves only have a "sale license". If you ever want to buy a game you need to contract EA and bring a friggin big suitcase.
And it is generally very uncommon to trade licenses/rights freely. The "first sale doctrine" (Which we don't exactly have in Germany, but something similar enough to use that term) is an exception altogether made necessary by the fact that physical products were "the vessel" of the license in question. Always getting the actual rights holder/trader to agree would have been rather unpractical.
To apply this workaround now on the digital market makes, were the limitation of physical media doesn't exist, makes no sense.