Decided to test this on a laptop running Pop! OS 20.10. Issues was reproducible there as well. The comments left on the Steam community thread I commented on suggested the others encountering this bug also ran Ubuntu derivatives.
I confirm this behaviour.
Debian 10.10 here, with NVidia backports drivers and Beta client, to use Proton. When asked for an age verification, the day and year are correct, but when I click on the month menu, it stays on the screen... well, on EACH screen of my multiple desktop environment.

The only way to get rid of this menu is to close Steam.
I'm seeing this on Arch Linux as well. Both current beta and stable are affected.
I'm also just passing here to confirm this. I'm using Neon 5.22 and the issue also occurs.

I'm experiencing this problem as well, typically when entering the month of my birth into a dropdown list to watch an R-Rated game trailer. I'm running Manjaro (5.10.68.1 kernel), Plasma 5.22.5 desktop. Lenovo AMD Ryzen 7 4750U laptop with Integrated Renior graphics. I've seen this problem for at least the past 6 weeks. Thanks for looking into it.
Nothing extracted yet.
Your system information
Please describe your issue in as much detail as possible:
When viewing my discovery cue a page comes up asking for age verification. I pick my age and instead of the dialogue box disappearing it remains on screen even if I exit Steam. A reboot is required to remove it.
Oddly, if I control-click an item to open a new window BEFORE I interact with any age pickers I can get rid of this rogue dialogue box by just closing the window. The former Steam window is unaffected
Steps for reproducing this issue: