Finally I found the problem and solved it so I think I'll share it 'cause it might help someone in the same shoes.
The problem was that the plugin stores the files on the server in the following way. It creates a hash which will be the name of the file in the file system and creates a folder with the name of the first letter of the hash. So for example s9lzdrefscjhmrt5mp47q34rtrvdo456 will be the file name which will be stored in a directory called s.
When I moved my files from the old server I zipped the attachments folder and during that the uppercase and lowercase folder of the same letters were merged into one folder, for an unknown reason. So for example I had a folder called A and a and by the end of the zipping process I had only an a folder which contained the files from a and A. This caused that after uploading them to the new server I got a Missing file error for every file which hash' started with A.
After I have restored the original folder structure and folder contents, the problem is gone an I can open every file on my server.
ntozier - thank you for your help!