What happened to the shared stash? It doesn't seem to exist anymore!

It is still there, at least for me, press the top left button when you in your stash.

It's not going to disappear, there'd be no point.

I've lost my stash. I was saving in the cloud of GOG. Had terrible connections at that time. I assume that bad connection screwed up my stash. Since then I removed the game from saving in the cloud.

Cloud is a good solution. No doubt about it. But when your connection to the cloud is not stable due to provider problems, it corrupt your savings. I assume that there are save-files for every character and the shared stash is an extra file.

All assumptions and POV, but nothing else makes sense. GOG is NOT at fault!

Probably. While it's advised not to use Steam's cloud saving, GOG's is usually okay.

Gog Cloud is no real Cloud saving like Steam, you still have your local Savefiles and the are synched with the cloud. Could still be, the cloudfile with same name was marked newer as local file and thus overwrote it ... still the cloud ist quite good for me, though it doesnt have too much space.

I guess this is what happened.
My bad connection corrupted the files written in the cloud. When I reconnected the files in the cloud were the ones that should have been better because of the dates. Thus the corrupted files overwrote the local files.
I did NOT loose the character files. I only lost the stash.

As I mentioned before: the cloud is good. It did not work because of my personal circumstance. Thus I only recommend to NOT use cloud, if the connection used is NOT stable.