Agreed, if he's still drinking every day he's an active alcoholic, not in recovery.
Is it normal? In the sense that some alcoholics live in denial of their problem and keep drinking anyway, yes. That doesn't make it acceptable though.
Ultimatums are a tricky thing....because at the end of the day, only he can make the decision to get sober. No amount of intervention, rehab, or whatever you can dish out can force an alcoholic to quit. It has to be a decision that comes from within.
On the other hand, If you have simply had enough, an ultimatum for your departure from the relationship could be a boundary you have to set. If you are unhappy in the relationship and he's unwilling to change, taking care of yourself and making change for the better may be the only option.
If he's in AA does he have a sponsor and work the steps? Or just go to the meetings? Have you discussed openly with him your issues with his drinking?