They can't be concussions from shaking your head side to side, so no, your brain isn't hitting the side of your skull. You haven't fully healed yet from your concussion, and these shakes are aggravating the initial injury in some way, in my opinion. It could be related to your head or neck, or both. With that being said, your doctor obviously thinks these are harmless, or he wouldn't have you doing them. If I were you I'd stop them, but other people, like your doctor, might advise otherwise. Its really your call to make.

I had a medical professional of some sort tell me to do the exact same head shaking exercises, and I really didnt like it so I didnt go back. I told my doctor that it wasn't worth the risk of prolonging my recovery. I don't think he agreed (since this head sensitivity is pretty rare amongst concussion patients and not yet widely recognized as an actual thing).

I also have the sensitivity to bumps, and usually after things flare up, they revert back to what *my* normal situation is after anywhere from a couple days to a month. I've been told that these aren't doing any damage, but just causing setbacks of some sort, as you put it.