Is there any way to fix that and use multiple files with the same name inside my Qt project? Does it perhaps have something to do with the fact, that I am compiling into a bin dir inside my repository?
Not with shadow building, no. When you generate a bunch of object files in the same directory, the object is named after the source file. So the identically named source's object file will overwrite the previous one (in order of building); it may even happen only partially if you're building on multiple cores.
It's a known pitfall and you should name your files better as people have already suggested.