Nexus 5X: How can I use custom ringtones /alarms without installing additional apps?
I finally found a solution: I deleted all files in /Alarms and /Ringtones and rebooted. Then I copied just one mp3-file to each folder with the built-in file explorer (I guess it does not matter which app you use or if you do it via pc/usb). This mp3-file could be selected in alarm and ringtone settings.

My first idea was that one of the files that were in /Alarms and /Ringtones before was damaged or in a not supported format. So I copied back the other files. To my surprise it is still working and I can also select the other files. So I don't really know what caused the problem, but this work-around solved it.

So if you have a similar problem, it may help to delete all files in that folders (make a backup somewhere else), reboot and then copy the files back.