I also have a painted concrete shower floor. It was already painted when we moved in, and has since gotten hard to clean and the paint is peeling. I would like to put tile (or something) over the floor that would look better, rather than just find the right paint again.

Yes you can but will need to get the paint off first and then rough up the floor a bit. Use a good grade masonry tile setting compound, not Acrylpro or something like that. You may need to cut some tiles to get the proper pitch to the drain.

You will need to do more than just grind off the paint. Soaps, shampoo, conditioner and body oils have soaked into the concrete. You'll have to do some heavy duty cleaning.

It also sounds like this is a make-shift shower and not one built from scratch with membrane etc. Has it worked OK all these years? Any leaks out of the shower space?

Jaz