There are special dowels for drywall.
Maybe not there where you are but there are.
Yes, but those don’t work for heavy applications like this. They usually have a weight rating, one well under a full rack of clothes… Just put it into the wall studs, it’s easy enough.
Well please excuse my German stubbornness… The drywall anchors (thanks @stevehobbes@lemm.ee for the word) I know and often installed were certified up to 40 kilogram - per anchor (single layer drywall, one anchor per meter as a rule of thumb). A rack full of stuff like in the picture won’t have more than 50 kilogram load (static load - please tell your kids not to climb onto the shelves) so two such anchors would have handled that easily, and there’s plenty of room to install more than two.
I understand that other parts of the world live up to other standards.
There are ways to attach shelves to drywall alone, that can hold quite many kilograms. How were the shelves attached and how much stuff have you put on there?
They used cheap plastic drywall anchors, the kind that split into two small “half-rods” (for lack of a better term) when you screw in. The problem is the anchors are plastic and the plastic bends pretty easily, so the load isn’t really distributed much at all. If you look at the holes near the bottom of where the shelf was that’ll give you an idea of how little they spread out.
As for how much stuff, a handful of boxes with things I access semi-frequently, mostly game controllers, network cables, extension cords, so on. No books or anything. Far less than I’ve put up in other places I lived at actually. More to the point, far less than what the previous occupant had up there (we helped move out her stuff).
They can be mounted on drywall… with anchors
My parents bought a house a couple years ago, and the previous owners thought it would be okay to put a wall in by screwing 2x4s into the existing drywall of the adjacent walls and putting the sheets up on either side of the 2x4.
They also thought that nobody would notice the fresh paint and fresh plywood used to cover up massive amounts of mold, and when it was discovered after the first day they shrugged and went “oh we had no idea” and a judge believed them.
This happened to me when I bought my first house. Came home to clothes all over the place due to cheap plastic anchors. So annoying to fix, and a sign of other problems to come. Lazy contractors gonna lazy