I am new here and have little experience in repairing bicycles. I have an about 15-year-old bicycle with an 8 gear Shimano Nexus internally geared hub (SG-8C31). Unfortunately, the back wheel is not tight and stationary, but can be pushed a few mm to the sides. I wonder whether this is a capital damage or if it can be repaired. I don't know what parts are responsible for the wobbliness nor how they are named so it is hard to find information.
I think the ring on the second image is the cause of the problem. If I tighten it, the wheel stops wobbling. If I tighten it too hard, it also stops spinning. Are there things to consider when thightening this or is it just about finding the sweet spot between wobbly wheel and fix wheel?