Having this issue as well. Basically the same as #3301. Happens on both the Steam Client Beta, and not (on the latest versions available today, May 8th 2020).
Running Ubuntu 18.04 with Cinnamon.
On my Linux Mint Cinnamon it's like this since February 2018. See #5379, which sadly never got fixed. I have to use a desktop shortcut with "killall steam", or my shutdown takes forever.
The comments on https://github.com/ValveSoftware/steam-for-linux/issues/5379 suggest that this is specific to Cinnamon DE - or more likely its Window Manager, a fork of GNOME3's Window Manager according to wikipedia? Are you using Wayland at all, or is this on XOrg?
I'm on Manjaro with KDE Plasma. I don't think there's much in common with GNOME3's Window Manager?
In any case, though, Steam is the only application showcasing this behavior (as far as I know). I truly wonder what the reason for this oddity is.
Your system information
Please describe your issue in as much detail as possible:
When trying to shut down or reboot my computer using menu options, only Steam gets closed. A second attempt will then actually work.
This happens across all of my computers (various different hardware) and across all distributions I use.
Basically the same as #3301 - which never should have been closed, but seems to have been forgotten.
Shutting down via console "shutdown -h 0" works just fine, btw. Not really sure what the difference is.
Steps for reproducing this issue: