You should decide who is the owner of the files there. And when you tell you want to put backups of files there, you should decide who is making the backups.
When it is root that is doing this for the users (maybe only the user <user>), then root should own the mount point and do the copy while keeping ownership and permissions (e.g. cp -a). I do not know what Dolphin “in superuser mode” exactly does. Try to avoid using such GUI programs as root. And it should definitely be mounted elsewhere.
When it is the user <user> that does make his/her own backups, the mount point can be used and Dolphin can be used also (I would prefer a backup script though). But then user <user> must own everything there. And that is where the chown is for.
When you think that is needed. But we can only use <user> because we do not know what it really is and you have to translate everything.
Please copy/paste as I explained above. We want to see things, not your description about what you think you saw.
You probably did this as <user> and not as root, but we can not tell because you did not show us what you did. Of course it should be done as root, because, as tyou have found out already, <user> can not do very much which files it does not won.
You mean you have three partitions there (no sdb1?) That is OK.
As said above, when it is user <user> that makes the backup (and restores), that is fine. <user> plugs in the device and the desktop will offer him/her to use the file manager on it. Then <user> can drag to and from. All fine, But <user> must then be the owner of those directories.
And some advice.
Please try to understand how a multi-user / multi-session operating system like Unix/Linux functions. Users must be protected from each other. Thus concepts like ownership (by user and by group) and permission granted to owner, users in the same group and other (the world) to files. There is plenty of course material on the internet.
Also try to understand that stories like “I did this” and “I can not write” is of not much use. The Linux system does not know that “I” (you), it only knows about users defined on the system. So it is user <user> that does something and user root may do something different, but it is never “I”.