I managed to fix this by using
gnome-disks to format the ext4 partition instead of initially using GParted, as well as ensuring the Dropbox directory was placed at a 2-level depth of the mount point of the partition. On 64-bit Ubuntu 18.04.1 LTS with Dropbox v60.4.107.
The full storyline:
To begin with I created my dedicated ext4 with GParted which also was used to resize the old partition to allow room for the new ext4.
Then I tried to ensure that my setup fulfilled everything described by Florian's answer, but that did not solve my problem.
Then after many tries at combining different solutions, I decided to erase the whole partition and reformat it to ext4 with Ubuntu's native disk manager (just called Disks or
gnome-disks) leading to Dropbox accepting the partition as ext4!
I used the following line in
/etc/fstab to mount the partition:
UUID=ext4_partition_UUID /media/dropbox ext4 defaults 0 2
ext4_partition_UUID represents the UUID found with
ls -l /dev/disk/by-uuid/)
Notice that I do not specify the
user_xattr option here.
My Dropbox folder is now located at
/media/dropbox/data/Dropbox - but did not check if the depth of 2 was really necessary.
It seems something went wrong when formatting the ext4 partition with GParted instead of the native software - no idea why or what the difference between them would be. If anyone knows I would be happy to learn more about it.