The exact mechanism you describe for how suction cups work is how the rice bowl work. Instead of the bowl being flexible, though, it's his body (skin and muscles) that are providing the change in volume necessary for the suction. So, instead of the suction cup creating the volume change, it's the surface the suction cup (the bowl) is attached to, Mr. Zhang's very impressive abs.
All you need for suction is for the air pressure inside the bowl to be less than the air outside the bowl. I presume that Mr. Zhang accomplishes this by filling as much of the bowl as he can with his own skin and muscle, and then uses impressive abdominal strength to pull that skin and muscle back, increasing the volume accessible to the air in the bowl and thus decreasing its pressure.
Also note that Mr. Zhang has a strap that helps hold the bowl to his gut. I am not implying that this strap is being used to actually hold the bowl onto his gut, but probably more to keep the bowl positioned well enough so that the suction is not broken.