Issue: double categories

I don't know what you mean. I only see tags at the bottom of the page. No categories.

There are two categories "weltkindertag" in the category list

I don't see a category list. The only thing in the usual box for me is:
"Schlagworte: weltkindertag"

In that case, weltkindertag is a tag, which is completely different from a category. It is possible to see "weltkindertag" three times at the bottom of the page, but under different headings. That means you have three different kinds of content named "weltkindertag", and you have added the current page to all of them:
- A category called "weltkindertag"
- A tag called "weltkindertag"
- A private feed called "weltkindertag"

If you see "weltkindertag" multiple times under the same heading (e.g. twice under feeds or twice under categories), then that means you have created a name conflict for "weltkindertag", and you added the current page to both conflicts. You should use the Maintenance > URL Conflicts tool in the Wiki CP to remove the duplicates if this is the case.

Have a look:
http://geislingen.net/wiki/Kategorie

There are two entries "weltkindertag"

This is rare, but same as: https://www.vaultwiki.org/issues/4312/
I have now been able to isolate the problem.