Yes, Drinking Societies do exist, and there are quite a few of them in Oxford...
Although there is quite a spectrum of clubs out there, the one thing they tend to have in common is that most are by invitation only. The standard procedure is for potential members to be proposed, and they will get invited as long as they're not blackballed by any existing member.
First, there are the very select university-wide ones, like the Bullingdon. About 12 members, mostly Old Etonian, and titled. They get very drunk and destroy a lot of things, which they later have to pay for.
A college will normally have more than one society, and they vaguely resemble either of the following:
The College-based dining clubs, which can be very old, and are a posher version of the standard drinking societies. About 10 to 15 members, and they will have club ties (possibly jackets), and tend to be dominated by rahs and public school types. So invitation will depend on knowing the right people. I think they just go out for expensive dinners and drink lots of wine.
Then the pure drinking societies tend to be newer, less exclusive, but much more fun. Their activities will revolve largely around consuming ridiculous quantities of alcohol in an evening. These clubs tend to be dominated by laddish, rugby types, and notorious members of college. And most women's drinking societies tend to be quite similar - rowdy and fun.