Hi @UncleNo.1, unfortunately I am also using an RPM package, but I do have Brave installed :-
I am able to do it properly. According to the screenshot which you have provided :-
It does seem as a bug. But try to scroll down the settings page and then find for an appearance section.
I am using Fedora, and if I switch to light/dark mode, still I have to manually change the theme in installed apps.
May I know the version of Brave you currently have?
Nope not their see this and this.
But you are at least getting an option to change in my case appearance tab is broken However in my desktop chrome and microsoft edge are able to detect dark mode very properly but IDk what's the issue with brave.
It is the latest version. Thank you for the video as it solved many of my questions. You may try to re-install Brave once again and check if the problem gets solved 
My data will loss or not?
To sync with your fresh install of brave, do the following steps
You forgot to mention following steps 
go to settings, then click on sync option
over there, click on start a new sync chain. it will provide you a 24 word code.
Nope I don't want to sync with cloud so better I will lose my data but I won't sync.
no no it will not save in cloud
So where will it save
Your data is already saved in their cloud or your own device, which on entering the 24 word code, will back up automatically.
Done reinstalling however I backed up that folder but only bookmarks are back IDK why but I am OK, and after reinstalling nothing fixed.
Can I try installing from software store once?
Try installing from snap. Yes software store. install it from there
No snap one also same problem.
I get this but all the things are broken too much broken 
What an annoying bug, isn't it 
