I have found the cause of the problem on my site.
Somehow, I was not able to remove the wpautop filter (not with plugins or any PHP code), and the wrapping <p>-Tag for the Upload field created the Problem. The ajax-file-upload.js-Script was relying on .parent tags, which now gave the wrong results.
I modified the JavaScript to encounter for the <p>-tag, I hope to find a way to disable these annoying tags, but this works for now.
