I know lots of people who can put away 5-6 drinks and not be classified as alcoholics, but what you're discussing here is how YOU feel about his drinking. I would suggest trying an Al Anon meeting in your area. My husband does not even drink that much but I tend to classify him as an alcoholic because he fits the profile in more ways than just his drinking. He's a binge drinker and can't stop many times once he starts and he makes poor decisions when he drinks that affect our family.
You seem to have a lot of anxiety over his drinking, maybe exploring where this stems from and reading a few books on codependency might help you. There's certainly nothing wrong with a little self exploration. If your BF is a good man, treats you well, abides by society's laws, is responsible with his drinking, and is responsible for himself and his own well-being, then maybe the issue is something more than just the drinking. I know I had a lot of insecurities around my husband's drinking when we were dating because my dad was an alcoholic and I projected my fears about what happened in my childhood into my young adult life. I would judge people for how much they drank and then put myself up on a 'better than thou' pedestal because I stopped at 2 beers. Through Al Anon, I've learned that I need to check myself first before I start judging everyone else, and that includes my husband. Hang in there, keep coming back, and ask away! There's lots of great support here.