TL;DR: Perform a clean re-install of Visual Studio
After wasting several hours, I still couldn't fix it for Visual Studio 2017. I then installed Visual Studio 2019 PREVIEW and all of a sudden, IntelliSense displays me the members of STL-classes again (which it doesn't with Visual Studio 2017).
So, my guess is that there might be also something wrong with Visual Studio itself (maybe something in the cache directory or in general something on your PC which is not directly related to a specific solution) which can be solved by a clean and complete re-installation of Visual Studio. I know, this is a stupid "solution", but in my case, only a fresh Visual Studio (2019) installation had an effect.
As already mentioned, in my case, only STL classes were affected. IntelliSense wouldn't display their members which is odd. I thought, it could maybe have something to do with precompiled headers. Somewhere I read that the STL and the project should be on the same drive and putting them onto the same should solve the problem. But none of these routes led to success.