So, after thinking out loud in the Reddit group as well, the solution came as the following:
Running mtp-detect with the Kindle connected, as a user, resulted in the following:
libmtp version: 1.1.22
Listing raw device(s)
Device 0 (VID=1949 and PID=9981) is a Amazon Kindle Scribe 32GB.
Found 1 device(s):
Amazon: Kindle Scribe 32GB (1949:9981) @ bus 1, dev 9
Attempting to connect device(s)
libusb_open() failed!: Permission denied
LIBMTP PANIC: Unable to initialize device
Unable to open raw device 0
OK.
Running the same command as root gave a much better result:
libmtp version: 1.1.22
Listing raw device(s)
Device 0 (VID=1949 and PID=9981) is a Amazon Kindle Scribe 32GB.
Found 1 device(s):
Amazon: Kindle Scribe 32GB (1949:9981) @ bus 1, dev 25
Attempting to connect device(s)
USB low-level info:
bcdUSB: 512
bDeviceClass: 0
bDeviceSubClass: 0
bDeviceProtocol: 0
idVendor: 1949
idProduct: 9981
IN endpoint maxpacket: 512 bytes
OUT endpoint maxpacket: 512 bytes
Raw device info:
Bus location: 1
Device number: 25
Device entry info:
Vendor: Amazon
Vendor id: 0x1949
Product: Kindle Scribe 32GB
Product id: 0x9981
Device flags: 0x18008106
Configuration 0, interface 0, altsetting 0:
Interface description contains the string "MTP"
Device recognized as MTP, no further probing.
Device info:
Manufacturer: Amazon
Model: Kindle Scribe
[...]
This confirmed it is a permission issue. So I created the file /etc/udev/rules.d/99-kindle-scribe.rules with the following content:
SUBSYSTEM=="usb", ATTRS{idVendor}=="1949", ATTRS{idProduct}=="9981", MODE:="0666"
Reloaded the rules with:
# udevadm control --reload-rules
And Calibre can now see and manage the Kindle Scribe!
Thinking out loud in public really helps.