My autocrlf is equal to true. In my cygwin-shell git status gives me a correct list of all my changes.
In Git Bash git status says I modified all files in the project. I also see this in Git GUI and the Changes-tab in IntelliJ.
How is this possible, and more importantly, how can I fix it?
Cygwin Git "sees the world" as if it runs on a POSIX platform—thanks to the emulation provided by Cygwin. Contrary to this, Git for Windows is a native Windows program which does not use any emulation and tries to be as close to Windows standards (however idiotic) as possible. What this leads to is that for Cygwin Git, LF is the standard EOL character, while for Git for Windows, the native EOL sequence is CRLF. So both tools see the world differently, and that explains what you observe. Please read this recent thread for more info (and especially this message).
In either case, consider setting core.autocrlf to false anyway to avoid headaches bound to this "magic". I'm using GfW solely, and in the end switched that setting to false (it defaults to true) for good.
You might also find the extremely well commented .gitattributes file from the Mono project to be interesting to study.