I stopped to help a drunk the other morning. I thought he was a tramp. He stank to high heaven, was poorly dressed, seemed to have his ID and letters in a carrier bag and had stitched his own wounds on his face. He was able to fall off a wall he was sitting on. I managed to coax him off the wall and steered him towards the park, where at least if he fell over on grass he would be less likely to hurt himself.
He walked very slow, but didn't sway or stagger. He was teary. I asked him how long he had been sleeping rough. He asked me what day it was, and replied one day. I told him 'rubbish, I saw you the night before as well', he had lost track of the day although he was obsessive about checking the time (counting down the minutes until a public toilet opened).
I spent some time with him, it turned out he has a flat, but there was no way he would be able to walk the distance in the state he was in. He told me how ashamed he is of drinking so much, and how he's messed up his life.
Noone chooses to live like that. Alcoholism is a mental obsession that causes a physical compulsion to drink. So yes, an illness I'd say it is.