The box itself is needed to pull this off, the box must shoulder against the wall to stop any flexing. Typically, you'd install picture frames of 2x4's on the wall, screwed (preferably Lag Screwed) into the studs & floor joists. I wouldn't go any wider than his 500mm (his look shorter than the 600mm kitchen counter). This isn't good for a structure not made for it & the boxes don't like it much either.
Then, you'd slip over your 3/4" plywood boxes (glued & screwed) & screw them to the picture frame all of the way around. And, screw the boxes to each other in 2 rows, like what he did at the very top panel. You'd actually build & connect the boxes first & dry fit them against the wall to then just pencil trace inside of the boxes for all of the needed picture frames that would go onto the wall first.
Then, you'd skin over your boxes & their unsightly screws with 1/4" plywood panels for concealment. Finally, as he did, you attach face frames to beautify all of the plywood edges.
More traditional box stairs are much better & more versatile storage solutions. Since, they can be any width, don't need additional framing & the supporting boxes can be drawers, cabinets or a full closet or armoire at the back end (if you put a TV or shelf on the side) or back side.