e.g. If I log into bspwm instead of plasma:
This is not a major problem and not limited to EnOS (had the same problem on Manjaro), but it is weird. Any ideas? Or an explanation?
It smells like something to do with your extensions or privacy settings rather than the Linux desktop.
If you log out but back into the same desktop, does it work then?
Nah, happens on all browsers and I run uBlock only, the other extensions I use are not relevant:
Super History & Cache Cleaner : version 1_2_7
Web Store : version 0_2
Plasma Integration : version 1_7_6
uBlock Origin : version 1_29_2
Vimium : version 1_66
Notifier for Gmail™ : version 0_9_6
Tampermonkey : version 4_10
Dark Reader : version 4_9_19
I don't care about cookies : version 3_2_1
Stylish - Custom themes for any website : version 2_0_9
Bukubrow : version 5_0_2_0
TweetDeck by Twitter : version 3_10
Tab Session Manager : version 6_2_0
Vivaldi Picture-In-Picture : version 1_0
Power Close : version 0_5_2
CryptoTokenExtension : version 0_9_74
Cloud Print : version 0_1
Chromium PDF Viewer : version 1
BetterTweetDeck : version 3_9_15
Pace4Chrome : version 2_0
Vivaldi : version 1_5
Google Hangouts : version 1_3_15
Bitwarden - Free Password Manager : version 1_45_0
WebRTC Network Limiter : version 0_2_1_3
Grammar and Spell Checker - LanguageTool : version 3_1_10
Hover Zoom+ : version 1_0_155
Chrome Media Router : version 8520_615_0_5
Are you sure about that? 
It does sound like the browser is clearing site data - I have mine set to do that on close, so I have to log in to sites every time I restart the browser.
In my book that’s the way it should be. Now that said when you logout or restart the system browsers and email clients are items that should be closed first. Now like @Beardedgeek72 asked if you log back into the same desktop does the behavior still happen?
It has to be clicked to run it, so no, it is not relevant, I installed it for a reason that completely escapes me now 
Of course it works if I stay on the same desktop, only occurs if I swap desktops, it is all in the title
Quick demo, on plasma:
Logging out, see you on the bspwm side.
Logged out of the forum and tweetdeck.
If cookies are removed - you are effectively removing active logins.
No, that extension does not do that:
Remove cookie warnings from almost all websites! The EU regulations require that any website using tracking cookies must get…
As you describe - you are only logged out of everything when you switch desktop environment.
This implies logging out of the current session.
If you are using an app which 'caches* your browser tmps to memory and only writes to disk on exist - you could have a flawed config with respect to the actual implementation.
I have used, in the past:
1 community/profile-sync-daemon 6.40-1 (24.2 KiB 47.0 KiB)
Symlinks and syncs browser profile dirs to RAM
But don’t use it anymore, due to constant crashes. This also happened on Manjaro (I just put up with it).
And it is not just me:
But that is all I can find.