Acknowledging that they are obese is stating a fact. Claiming that they wouldn't or couldn't take good care of their dog because of their obesity is fat shaming. They're not 600+lbs-can't-leave-the-bed obese.
Exactly. People who fat shame don't actually want people to improve -- they just want a target for their hate. I see no reason to disparage the people in the picture, nor to think they won't properly care for their dog.
So obviously that means he couldn't possibly ever exercise, because people who lose weight just wake up suddenly skinny and are never still fat at any of the in-between points during which they still get constantly shit on for existing in a shape society disapproves of.