Enabling the permission in Android settings helped. Must way, not very user friendly.
One strange thing though, which is why I assumed that these permissions was already granted. Opera had no problems displaying a listing of my external directory (/data/media/0) as to allow me to select the folder for my downloaded files. The read external permission is in the same group as the write external. If one permission in a specific group is granted, all permissions within the same group is automatically granted by Android, at least according to their documentation. So if Opera was able to read from external, it should also have been able to write e.g. save my download.