First up, if the furniture isn't that old, first check if it is still under warranty because there's no point trying it yourself if the seller still covers faults. And this is clearly not damage from use, this was a poorly seasoned piece of wood that should never have been used.
Also check all the rest of the drawers and any other pieces in this set.
If it were me, I'd consider that this is the side of the drawer, where function is more important than looks.
If warranty is past, then I'd find some wood of the same thickness as the drawer side - about 10mm at a guess. Then I would cut this shape out of the side of your drawer, trying for straight lines where possible.
Measure the resulting space, and cut a new patch piece slightly larger - you want a tight fit, something you need to push, but not hammer into place. Once the patch is sized, add the start of the roundover on top because it will be harder to do when in place.

To install, glue the three mating surfaces with wood glue, push the patch into place, clamp it up while wiping up any squeeze-out.
Yes - the top dovetail on that side will be purely decorative and not adding any strength because you're gluing end grain. It might help to add some strength by gluing in some coving or quarter-round or any convent shape in that inside-corner. It will slightly intrude on the drawer space, but only 20 cc or so.
Another option is to extend your patch piece all the way to the rear of the drawer - there's a non-zero chance that there is still tension in this board and it may crack further when you cut it.
When all the glue is truely dry, probably a week or so, then add a light stain to match the rest of the drawer. Doesn't have to be perfect, as you can see the existing drawer was stained too.