Thank you, I have already read the documentation and all permissions are set correctly.
But meanwhile I found the issue by my own.
Maybe it will help someone who has the same issue:
In my case the default value for session.save_path was not included in open_basedir.
So I created a new folder with permission 770 inside open_basedir (because of security reasons because default session.save_path was outside {WEBSPACEROOT}) and that solved both issues.