There is absolutely a stigma against obesity in the US. The country’s fixation on weight is well-documented and the last decade of “body-positivity” has done little to change it.
That said, the stigma against obesity is absolutely less severe than the stigma against drug addiction. Personally, I think that’s appropriate.
One can be a healthy addict (i know very healthy people who cannot stop coffee without withdrawal), one cannot be a healthy obese person.
The problem in this comparison is that addiction has a very broad scale (e.g. coffee to crack) similar to body weight. Obesity already entails the person well over-weight (in terms of drug use that'd be "daily crack use" or smth).
I personally would rather be in a relation with a coffee addict than someone suffering from obesity.
That said, the stigma against obesity is absolutely less severe than the stigma against drug addiction. Personally, I think that’s appropriate.