You might try some method of steam, I detail a huge gym and have been researching the best way to clean grout and shower stalls and steam they say is the only way to remove build up like this. google "hand held, steam cleaners".
If the glazed has been worn off, then the bottom is porous and the stains seep down in the concrete. May want to contact a professional and see what they suggest.
I agree with John about Barkeeper's Friend. I've also successfully used Oxy Clean (a paste made from the powder) or Ceramabryte, which is a product used to clean glass stove tops. I've found it works on other things, too, without scratching.
I would try a Mr.Clean spounge pad first. If that didn't work I would get Sno-bowl toilet bowl cleaner and a brush after it. Let it sit on it for a while, scrub it and turn the shower on and rinse. try to avoid inhaling fumes. Open a window and wear rubber gloves. Have you tried CLR? that might work to.
You might want to try TSP. You can find it at Lowe's or other hardware stores. Sometimes Wal-Mart. Follow the directions. It worked on the pool bottom so at my wits end, I tried it on the shower floor drain and it took off the built up lime and soap scum. Depends on the sealant on the shower floor.
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