That's not a stain, that's patina!
Says the vintage holder guy.
If it's acrylic, then very hot water and scrubbing works. Unfortunately, the water will also soak into the end grain of the wood. There is the possibility of it warping the holder a little, or even cracking it if it dries too quickly.
If you really want to get it off, what I would do is try and soak it in very hot water and scrub with a toothbrush, stiff bristles, if you can find one these days. I've heard Pine-sol works as well after a soak. May damage the finish, though. I don't know. You could write to John Neal Books and ask. They know a lot of things.