This is a long-time bug that is well known. Without internet connection, you need to have given it a successful login credential so that it knows which account is to be in offline mode.
Closing as this is not a Linux-specific issue. Also, there have been some changes around offline mode in the last year and hopefully the behavior is improving.
I can confirm that this bug was, in fact, fixed.
Nothing extracted yet.
When I'm run Steam when no internet connection i see that dialog:
![1]()
So, when i'm push Quit - anyway runs Offline-mode...
This bug also can be reproduced in windows version.