The reason the shower valve socket wrench fails - is because that is not the valve you have.
You have a Cartridge and a Cartridge removal tool is used for that.
Typically these units have a retaining ring that you remove first, then you might need to turn the cartridge unit a quarter turn and then pull straight out.
Your Cartridge looks to be an el-cheapo type (plastic) if it has been in there a while - you will have some fun removing it as they typically get stuck in the housing.
So these removal tools are normally a tube and nut assembly with a screw in the center - you tighten in your center screw - and then would turn the outer nut which would apply force in pulling the cartridge straight out.
As noted in comments:
Cartridge replacement tools such as this Danco Off Brand used for Moen Valve Assemblies to pull cartridges are Manufacturer specific.
Be sure to take note of your manufacturer and model of unit. Some manufacturers warrant their cartridges and valves - check with the manufacturer for details regarding yours.