OSX - File Sharing - Permission problems
Buscar, thanks a lot and you were right with the ACL issue.
I would like to "vote up" your answer, but I can't do that if it's just a comment? At least I didn't figure out how.

In any case: I would recommend to look into this explanation and way forward:
https://gist.github.com/nelstrom/4988643

What needs to be done is this:
In OSX High Sierra:


Create a Share, e.g. /Users/admin/share. 
Create a Group, e.g. "sharegroup", and share /Users/admin/share to that groupo and give read&amp;write permissions.
Add users to the group, that are supposed to have access to the folder  
run the following command from the github link above: 

chmod -R +a "group:sharegroup allow list,add_file,search,add_subdirectory,delete_child,readattr,writeattr,readextattr,writeextattr,readsecurity,file_inherit,directory_inherit" /Users/admin/share


And after this small command, I could use the mac Mini as a "normal" file server. Several users could access the shared folder(s).

I did not (have to) run the other command that was stated in the github link:  

 chown -R demo:sharegroup /Users/admin/share


I wasn't sure what it does.

If someone else should read this post and should (same as I) have an issue in understanding what ACL's are and what this setting change means, I personally found this forum thread helpful (but take care since the "chmod command" is faulty, when being copied and pasted):

https://discussions.apple.com/thread/7566517?page=2