How to clean this grey stain from the bathtub

It looks like you might be able to do this pretty easily.

To remove the stopper, simply unscrew the top of the handle of the drain stopper, then unscrew the underlying screw in the shaft to pull out the drain cover completely. Then, you'll have the cradle-thing that holds the stopper. You might be able to leave that in place (I grew up in a house that had the stoppers taken out but the "cradles" left in place, and two women with long hair would share the bath. We took turns cleaning out the drain) and place the mesh catcher over it. See this video for a demo (It should be region-free.)

You'll probably want to put down putty or some sort of long screw from the drain to the "cradle" in order to keep it in place.

Try using straight vinegar, right from a spray bottle. It may work better if it's heated first. Then try wiping with a used dryer sheet.

If that doesn't work, then you'll probably have to resort to something like CLR, at least for the initial cleaning.