Thanks for the lot of answers.
@drewDaeye: Meanwhile, I have copied my files with an USB stick. It took long, but it worked
FTP can be another way. I may give it a try later. Bluetooth is another option I was thinking on but it may be too slow to copy some gigabytes of data. I hardly ever used BT before.
@Flemur: There are 2 options I can choose from in my Android. One is media device (MTP) the other is camera (PTP). I have not seen any option for Mass Storage. However, if I remember well, my cellphone which has an older version of Android (4.0 - ICS, I think) supports USB Mass Storage as well. This is exatly the point why I am pretty disappointed. I did not expect any problem with a new Android device.
@Verändert2.0: I have already tried simple-mtpfs without success. I plan to give a go of go-mtpfs, too.
Let's go back to my first post for a moment. Do you have any idea what is the file ~/.mtpz-data used for? What if would try it with root privileges?
Unfortunately, I will have no time to deal with this issue before Monday evening.
I will be back.