Why are addicts their addiction while the mentally ill are not their disorder?

When I say, “I’m bipolar,” people often respond by saying, “you are not your disease.” I see people say the same to folks with cancer.

However, at twelve step meetings, everyone introduces themselves by saying, “I’m [name], and I’m an alcoholic [drug addict/sex and love addict, etc.]” It’s the same language that makes people say “you are not your disorder.” However, I haven’t heard people say, “you are not your addiction.”

I suppose people could say it and I just haven’t noticed. But the language of twelve step programs clearly identifies individuals with their addiction.

So, is it true that addicts are their addiction? If so, why are addicts their addiction while the mentally ill are not their disorder? If not, what’s with the twelve step language?