Protective shell as noise protection against neighbors

The Rigips boards are available in different colors and thicknesses, the blue ones as acoustic boards, for example. There are tables from the manufacturer for single and double cladding. I would look for a wall-filling shelving solution, clad at the back with double 12 or 15mm, and place it freestanding with a 5cm gap in front of the wall. Rental apartment = reversible.

yes, I have also learned that the construction must be installed floating. I also have great concerns that it will not be the solution. But I don't know what else to do.

I think most of the noise comes through the wall, since that is the weakest point of the construction, trampling over the floor. Since I don't hear the neighbors above and below me, I assume that the ceiling and floor are significantly better than the one side wall.

Stomping on the floor next door, but coming through the wall. Hmm. The question really is to what extent a secondary wall will bring a noticeably good improvement if the stomping next door is transmitted via the screed to the room wall. This, in turn, is firmly connected to the ceiling, floor, and other walls.

The alternative is first to talk to each other. Then you could consider whether the use of the affected room can be changed so that it becomes the adjacent room. Before incurring expenses, you might have the chance to place an existing bookshelf or cabinet against that wall. Ultimately, moving would be the last resort.

In principle, there are 13 (!) transmission paths for airborne sound transmission between 2 adjacent rooms. However, with your [Vorsatzschale] you only directly influence 5 of them. Whether you will achieve the desired effect with this is therefore at least questionable.