Unfortunately the odds of either a repair or removal without more damage are slim to none.

If the broken piece is in one piece and fits nicely on the break surface, try superglue. You pnly get one shot at it as the superglue will set very quickly but the porous surface at the break should adhere really well. At least it did when I glued my soapdish back together :)

If you repair it, (which I would as you have nothing to lose), please keep something...a folded towel maybe? on the tub floor underneath the dish for a few months. My tub chipped when my towel rail came off...

I'd score it with a carbide blade in an oscillating tool then break it up by gently tapping with a chisel. First take out the grout surrounding it, then see if there is enough room to slip the oscillating tool blade between the soap dish and the wall tile. If you can score from the back side it should be easy to break it up without damaging the tile.

Alternately take out both the dish and surrounding tiles and make it a "feature" by using a colored or patterned tile around the soap dish if you can't find matching replacement tiles. Some makers (eg American Olean) have replacement tiles that are still astonishingly good matches decades later for their standard tiles.

Hey @Avanti - which Sikaflex?

Superglue plus baking soda!

Sikaflex construction adhesive...