Okay, so apparently the culprit is the IntelliJ IDEA Project Creation Wizard.
If you create a new project within PyCharms and choose Django Project as desired Project type, it just works. You don't have to configure anything else.
To do this in IntelliJ IDEA, create a new Project, choose "Python Module" as type and check Django as technology. In the new project go to "File > Project Structure", navigate to "Facets", choose your Django Module on the right and set the "Settings:" option to point to the specific "settings.py" file.
After this configuration, everything should work as smooth as in PyCharms.
I consider this problem as a bug, as the Wizard creates the basic project, including a "settings.py" file, but doesn't add it in the project settings. Plus, you don't get any warning that such a strongly needed settings file is missing.