If someone responded to my statement by saying, "No. That is how you treat your problem," I would take it as a reminder that identifying as an alcoholic and a drug addict doesn't define my core problem. Instead, it signifies the way I've been dealing with or coping with an underlying issue in my life.
In this situation, my answer would be more introspective and focused on understanding the root cause of my addictive behaviors. I might respond with something like:
"My problem is that I have been using alcohol and drugs as a way to cope with deeper issues and emotions in my life. I need to address those underlying problems and find healthier ways to deal with them."