Do they sell you licenses like Steam or do you own the games after you buy them at Gamersgate?

Like anywhere, they sell you the license to use a game. Even on GOG (regardless of what GOG say in their blurb) or when buying retail, you don't actually own the game.

If you're of the mind that whatever's in the EULA is the way things are, then every copy of every game you "buy", whether it is from Steam, Gamersgate, GOG, or even a physical copy, is really just licensed. If you're of the mind that "buying" vs "licensed" is defined more by the actual characteristics of the transaction, such that tying a game to a service (like Steam) makes it licensed, while a no-strings-attached transaction (like GOG or a physical copy) constitutes an actual sale, then Gamersgate is in a bit of a gray area. Installing a game from Gamersgate requires using a downloader they provide, and once it's finished it decrypts the installation executable and installs the game; once this is finished the unencrypted executable is automatically deleted. If you want to reinstall the game then you'll need to use their downloader again (thus tying the game to Gamersgate like a service). However, it's easy to simply make a copy of the installer executable during the installation process, thus giving you all the installation files for future use, no strings attached. To complicate this a bit more, some of the games on Gamersgate also come with third-party DRM, requiring activation of the game after installation, thus tying the game to activation servers operated by someone other than Gamersgate. Sorry if that answer is a bit convoluted, but that's how it is.

You never buy the game. Buying it would mean that now you have the rights for it. You only buy a license to play the game.

So in the early 90s, I was buying only licenses after buying PC games at a store?

They can take my games away from my cold dead hands! ;-)

90s, 80s, 70s, 60s, before you were born, and even earlier times!

Legally speaking, yes.
Buying a game means acquiring the rights to it. Obviously, that would cost you more than the price of a single copy of a software.

As to your actual question, Gamersgate is somewhere between GOG and Steam in terms of restricting the use of the license they sell you.
That's because GG has no actual DRM system of its own (except for an easily bypassed encryption), it only uses the DRM the publisher wants to put on the game.
That can mean Steamworks, Securom (different versions) or (almost) nothing.

If you're willing to take the game companies' word for it, then yes. The actual history of the whole thing is a bit more involved. Initially software was regarded as a good just like any other physical copyrighted work (a book, an audio casset, a VHS tape, etc)- you were buying a copy of the work, but a copy that you did indeed own. However, some folks in the software industry noted that in order to run software one typically needed to copy portions of it to the hard drive and to RAM, and figured that these copies weren't authorized and hence to actually make use of the software people should need a license to do so, hence the birth of the EULA. Thus while you were still buying a copy of the software, to use it you needed to agree to the license that was included with the software. However, the copyright act in the US was amended to account for all this, explicitly allowing whatever copying needed to make use of a purchased program (17 USC section 117). However, by this point the idea that click-wrap EULAs could be foisted on everyone had become ingrained in the software industry, so despite there being to justifications for EULAs we still have them around. In any sensible world any EULA presented after money changes hands should be considered complete bullshit and completely unenforceable, but unfortunately the courts haven't been particularly sensible in quite a while and have thus upheld EULAs in several cases (although a few courts have struck them down in other cases).

Ah, this again.

You buy the experience, plain and simple. If the experience can be (to a degree) reproduced ad infinitum, you basically own that experience. If you only get to use it to a certain amount, then your experience is limited, therefore leased or paid-per-view.