Having studied and worked on this closely for over 30 years, I see several stages in the process of entering into, and recovering from, addiction.
Everyone is, to some degree, at risk of addiction. But some are much more at risk than others. And some are more at risk of one particular addiction (say alcohol) than others. (By the way, I believe that three separate things exist, obsession, behavior addiction, and substance addiction. But to meet your terms of the conversation, I'll talk only about substance addiction.)
Say someone realizes, "I am an alcoholic" and stops drinking. This person is still addicted, even if he never takes another drop. The intense desire is there.
Many people substitute one addiction for another. This can be healthy. Crack cocaine to alcohol is a step up. Alcohol to coffee is another step up. Then maybe the person becomes a wacky health food addict, but is still an addict.
If, in addition to dropping a particular addiction, the person goes seriously and deeply into rehab and a 12-step (or similar) program, a person can definitely, over time, loosen up those cravings, increase self-discipline, and *reduce* the addictive tendency. For almost everyone, this is as far as it goes.
So, for most people who have been addicted abusers of substances, it is best to stay with the 12-step language, "I'm me, and I'm an alcoholic. It has been 23 years, 2 months, and 5 days since I last took a drink." (You get the idea.)
But I have met a few people who find that, after a dozen or more years of 12-step, this is a limiting identity. With many years of deep spiritual and psychological work, it may be best to let go of the "addict" identity. I encourage such people (and many others) to follow The Artists Way by Julia Cameron, a program modelled on 12-step programs, but for recovering artistic creativity beyond addiction. And, at this point, it may be possible to say, "I'm Sid, a human being. Like all other people, I can be an addict. But it's been a long, long time."
On a personal note, I am fortunate enough not to have landed in any of the classic substance addictions. But I was addicted to foods that I was allergic to that made me quite sick. And now, following the steps described above, I can truly say that, looking deeply at my thoughts and feelings, the cravings are gone.