This is fixed in the next update (after New Years).
Today's update did not fix this issue, please reopen. I do not see any change in this regard.
Tooltop behaviour is exactly the same as before, in focus taking and appearance?
Indeed. Also, tooltip popups in Library section (e.g. when hovering over game achievement icons) started behaving peculiarly: when a tooltip is displayed, the focus switches quickly between the tooltip and the main window several times, then ends up on main window.
This is quite amusing when the window manager is set to display unfocused windows transparently - the main window flickers like crazy for about half a second.
Did yesterday's update make any difference?
Nope, I see same behavior as in my previous comment, even after update to the build from Jan 11.
How is the behaviour of this now, @ticho?
Still the same (using up to date client), see the screenshot taken few minutes ago: http://ticho.mine.nu/~ticho/compopups.png
Also, I just noticed that when I leave the popup visible and switch to a different virtual desktop and back without moving the mouse pointer, the popup is still there - another sign that the popup window does not have proper properties, since it should disappear.
I'm pretty sure this issue has been fixed? Could we maybe get some other re-tests, since this is a very old issue
Nope, it's been 9 years and this (albeit, not that huge) issue still persists.

Fedora 43 KDE Plasma 6.6.3 Wayland
While in newer versions the task manager "focus" indicator no longer flips back and forth with these menus open as in the image posted a few years ago, these menus still have some odd focus behavior.
The mouse exiting through the bottom of the dropdown leaves the dropdown on screen until you click elsewhere in the steam client.
Additionally, should you have a window open in front of the steam client and accidentally mouse over one of the top 'Store, Library' etc menus, the dropdown will open over the other front window, "sandwiching" the front window between the dropdown and the steam client, and will not disappear unless you refocus steam and click outside of the menu.
Edit: I also went into desktop mode on the Steam Deck to confirm it's happening there as well. The taskbar "focus indicator" highlighting still changes there as in the 2021 post above.
Nothing extracted yet.
The mouseover popups in Community section seem to behave as general top-level windows, instead of mere popup windows, so when they are displayed, focus goes from the Steam main window to the popup. This is incorrect, popups should behave same as for example popups in Store section.
I noticed this because I have set my window manager (KDE) to make unfocused windows somewhat transparent, and this is what happens: http://imageshack.us/photo/my-images/831/steampopup.png/