Those two locations are (normally) on the same disk, so it can’t be disk space trouble, so indeed I think it must be permission trouble. Of course backups will normally work - assuming you did a standard install - and so it seems likely that your manual creation of a directory has something to do with the trouble.
But. I checked on my machine, and the ownership and protection is like this, inside the container:
root@ubuntu-2gb-nbg1-1-app:/var/www/discourse# ls -ltrd /var/www/discourse/tmp/backups/default /var/www/discourse/public/backups/default
drwxr-xr-x 2 discourse www-data 4096 Jan 24 03:36 /var/www/discourse/tmp/backups/default
drwxr-xr-x 2 discourse www-data 4096 Jan 24 03:36 /var/www/discourse/public/backups/default
root@ubuntu-2gb-nbg1-1-app:/var/www/discourse# ls -nltrd /var/www/discourse/tmp/backups/default /var/www/discourse/public/backups/default
drwxr-xr-x 2 1000 33 4096 Jan 24 03:36 /var/www/discourse/tmp/backups/default
drwxr-xr-x 2 1000 33 4096 Jan 24 03:36 /var/www/discourse/public/backups/default
root@ubuntu-2gb-nbg1-1-app:/var/www/discourse# ls -nltrd /var/www/discourse/tmp/backups /var/www/discourse/public/backups
lrwxrwxrwx 1 0 0 19 Jan 10 15:26 /var/www/discourse/tmp/backups -> /shared/tmp/backups
lrwxrwxrwx 1 0 0 15 Jan 10 15:26 /var/www/discourse/public/backups -> /shared/backups
root@ubuntu-2gb-nbg1-1-app:/var/www/discourse# ls -nltrd /var/www/discourse/tmp /var/www/discourse/public
drwxr-xr-x 1 1000 1000 12288 Jan 10 15:31 /var/www/discourse/public
drwxr-xr-x 1 1000 33 4096 Jan 10 15:36 /var/www/discourse/tmp
root@ubuntu-2gb-nbg1-1-app:/var/www/discourse# ls -nltrd /var/www/discourse
drwxr-xr-x 1 1000 1000 4096 Jan 10 15:26 /var/www/discourse
root@ubuntu-2gb-nbg1-1-app:/var/www/discourse# ls -nltrd /shared/backups /shared/tmp/backups
drwxr-xr-x 3 1000 33 4096 Aug 15 2019 /shared/backups
drwxr-xr-x 3 1000 33 4096 Jan 10 15:36 /shared/tmp/backups
root@ubuntu-2gb-nbg1-1-app:/var/www/discourse# ls -nltrd /shared /shared/tmp
drwxr-xr-x 13 0 0 4096 Jan 27 2021 /shared
drwxr-xr-x 4 1000 33 4096 Jan 10 15:35 /shared/tmp
The 1000 user is “discourse” and the 33 group is “www-data” and all of this doesn’t give me a clue as to what your problem is. Sorry.