People are social creatures and one aspect of that is that we feel a natural, slight constant pressure to harmonize our mental states with the people around us. Diversity is good, of course, but we experience dissonance in outlook as an uncomfortable friction. When together, we always seek common ground to focus on.
When someone says "I don't drink"—an entirely valid and acceptable choice!—at a point in time where we've already established that someone else in the group does, it draws attention to that disharmony. It's not that not drinking is the bad thing, it's just the delta between that and the previously accepted reality that another in the group does. You've subtracted some territory from the common ground.
To get a sense of this, consider an imaginary scene:
A group of people are standing around trying to decide where to go out. The first few people suggest a place that is does not serve alcohol and that they are OK with that since they don't drink anyway. The next person mentions that they do drink.
In this case, it's now the drinker who looks flatfooted, at least as I imagine the scene. To me, then, this indicates that it's not being a non-drinker that's socially awkward, just that expressing a divergent preference that is.
Fortunately, most people have tact and can gracefully handle a little dissonance here regardless of which preference comes up first. Certainly, in the groups I frequent, we try very hard to make non-drinkers and drinkers feel comfortable and included regardless of who makes the first move.
The way this tact gets expressed is through plausible deniability. We express our preferences as vague or potential so that if a clashing preference comes up, it's easy to retreat from ours and get back to common ground.
That's why, at least in American speech, people say things like, "I could go for __." "I would be up for __." "__ sounds good." All of those are softly worded enough that they can be amended later to reach consensus without losing face.