I think it could be used in multiple situations. I immediately assumed he wasn't drinking because he's driving her home. Or he isn't drinking beer specifically because she doesn't like the breath it gives him, or is allergic to it. etc etc.
Curious. In English "drinking" in context refers to alcohol. If you said "Han drikker ikke for hennes skyld" would that make sense in that context, or would that just mean you'd be committing suicide via thirst? ;D