You can't tell from the picture how any of it was put together. More often than not the joinery was done with dowels and shelves could also be doweled. The corners of the legs could also have a dado where the shelf slips into and have a small nail from the underside.
Dados through at a 45 degree angle with matching 45 degree angles nipped from the shelf. Or a single pocket hole from the bottom of the shelf and a screw into the leg. Or both.
I've done it this way. takes careful planning/marking/cutting to get the right fit. the dowels will keep it together pretty well. I think I pin nailed from the side of the leg into the shelf.
The only answer I could come up with is the 45° dadoes.
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