I can copy text just fine, it will show up on the Clipboard Manager.

With images, however, most sources do not make it to the History, and it seems quite random.

The History does not remember these screenshots:

Some video games, fullscreen or windowed
Windows windows
MPC-HC when fullscreen
Internet browsers
Word and Excel
Discord
Audacity
Steam
etc.

The History does remember these screenshots:

Some video games
The Windows Notepad
Clipboard Fusion menus
MPC-HC when windowed
My email client

I've attached a couple of screenshots that might help.

Thank you, that did it!
I guess a 4K monitor has its disadvantages.
Weirdly, this setting had no value to begin with.