Every time I plug-in my USB flash drive I get this:

Device notifier provides me 2 actions.

If I hit "Open with File Manager" it refuses to mount removable media / open in Dolphin:

But if I open Dolphin myself and hit the appropriate icon:

The media is successfully mounts and there's opens another instance of Dolphin.

Why it doesn't work from Plasma Device Notifier? What can be the problem?

Interesting...
on my Neon User, I don't get that occurring so I had a look at my Removable Devices settings
I have all four tick boxes enabled
Maybe you have the Automatically mount removable media when attached disabled?
Hope that helps...

Unfortunately no, it doesn't help.

Apologies if you already saw this...
(but I just really dislike these kinda problems that should have simple solutions)
Given that you appear as a Gentoo user...
https://wiki.gentoo.org/wiki/KDE/Troubleshooting
Also, when you finally get remediation, please post what eventually worked...
Good luck!

Thanks, it worked.
Before it used to be automatic.