I'm asking this from a purely psychological and biological viewpoint, mainly why a drug addict is said to always remain one, even after sobering up.
Short answer, which is one of many:
One interesting hypothesis about this is the "reward hijacking" model. Essentially, when somebody becomes addicted their reward circuits become hypersensitive to drug-related cues and lack sensitivity to other rewards. Specifically, areas like their orbitofrontal cortex, which has to do with representing reward, and their ventral striatum, which has to do with dopamine and motivation, start to function strangely. So even when they replace the drug with new habits they still display heightened sensitivity to the drug. See Wrase et al, 2007 for one of many imaging studies on this. There are many neurobiological theories that explain why the reward sensitization occurs in response to addictive drugs.
The way this translates to function is that drugs appeal more and more to somebody with an addiction, while alternative habits appeal less and less. This contributes to the very high relapse rate. Interestingly, deep brain stimulation of the ventral striatum has helped curb alcohol cravings for some alcoholics. (See Knapp 2009 for rats, and Muller 2016 for a pilot study in humans.)
I hope this helps! The question you're asking is a very broad one, so you have a wide range of answers to explore.
Wrase, J., Schlagenhauf, F., Kienast, T., Wüstenberg, T., Bermpohl, F., Kahnt, T., ... & Heinz, A. (2007). Dysfunction of reward processing correlates with alcohol craving in detoxified alcoholics. Neuroimage, 35(2), 787-794.
Knapp, C. M., Tozier, L., Pak, A., Ciraulo, D. A., & Kornetsky, C. (2009). Deep brain stimulation of the nucleus accumbens reduces ethanol consumption in rats. Pharmacology Biochemistry and Behavior, 92(3), 474-479.
Müller, U. J., Sturm, V., Voges, J., Heinze, H. J., Galazky, I., Büntjen, L., ... & Bogerts, B. (2016). Nucleus Accumbens Deep Brain Stimulation for Alcohol Addiction–Safety and Clinical Long-term Results of a Pilot Trial. Pharmacopsychiatry, 49(04), 170-173.