How to remove the set key from shower faucet

I have a couple of questions for you. Why are you trying to remove this section of the mixing valve body? Are you planning on replacing the entire valve or trying to repair a faulty cartridge? The piece you are pointing to is normally part of the valve assembly and there should be no reason to remove it to change out a cartridge. If you are replacing the the valve, you may need to remove the whole body from the hot and cold feed lines and install a new body specific to the new fixture. This may not be a bad idea since it sounds like the old one may be stripped now. Another advantage will be updating to a scald guard style, and install a hammer arrester at the same time.

Most often when trying to remove old fittings like this, heat is necessary, lots of heat. Any internal parts, such as the cartridge, "O" rings etc must be removed so they will not melt. Mixing valves typically are difficult to solder/unsolder because they are large and require lots of heat. I usually use MAPP gas instead of propane.

The other thought I have is if the piece you are pointing to is two pieces, you will need two wrenches. One to hold the internal part and one to turn the larger external ring. If the internal part is already spinning, it may be damaged now.

As Steven says, hit it with some penetrating oil first (WD-40 should work for this). Give it some gentle whacks against a piece of scrap wood in all directions to help break any bonds. Then clamp on to the bottom square part and try to torque the nut on top with a socket. If you still can't get it, you may find it easier to replace the entire thing.