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This is electrical adjacent.
I removed the turn signal switch and saw something curious. I understand that the housing clamps to the steering column. On the switch side, there is a flat, crescent shaped, piece of metal that makes up one side of the clamp and the other side receives the screws.
When I took mine apart, there was a ring of duct tape that the crescent shaped piece would bite into. Obviously, not original to the car. It looks like it was wrapped repeatedly to make a larger surface to clap onto.

Is there a part that is missing?

I looked in the parts manual and didn’t see anything.

I suspect the tape was used as a shim -- possibly it was loose when tightened.
If that's the case you might be able to file the opposing sides near the screw
locations to allow a tighter fit. Just guessing a bit here.

As a bicycle nerd as well as a Pinzgauer nerd I keep a lot of bicycle inner tube material around for exactly this type of thing. I know that I have a section of standard thickness road bicycle (so pretty thin) inner tube rubber fitted between the crescent piece that you described and the steering column to prevent the turn switch from moving. The inner tube rubber is thick enough to take up any gap and adequately non-slip against the metal surfaces to keep the switch snuggly in place.

While I’m waiting for a response on my flasher inquiry, I took apart the switch to see if anything looked out of sorts. Everything looked as it should by my humble estimation.
I removed the layers of duct tape thinking there had to be a better way. I split open a small section of vacuum hose and put it on the crescent piece. This worked to hold the switch in place and it’s easily replaced if need be.
Also, now, the arm stays up or down. Previously, it was inconsistent.

On mine the turn signal assembly wants to rotate on the column, and I'm hesitant to overtighten the fasteners. Something grippy is likely a common mod folks do.