The best is if you could open the shower drain up and get the stuff out with your hands. Then advices what type of chemicals would help as well but later.

We had exactly same problem and no plunger and washing powder would get it out. Luckily our drain is not permanently mounted but locked with a screw so all it takes a little bit of time to undo it and clean it thoroughly inside. The loose hair is the biggest pain in the ass as it always clogs the drainage every few months.

If you can open it, clean it and then flush with strong chemicals to remove the rest of residue. Have a good look and good luck!