Hello,
Thank you for your replies!
I deactivated add-ons, restarted Firefox in safe mode and cleared the cache. Unfortunately it didn't help.
I installed Chrome and Vivaldi now and noticed the same strange behaviour there: Tipping the menu button on the touch screen opens the menu but clicking it with the mouse doesn't. Everything else works fine.
Only the Internet Explorer (11.0.9600.19036) opens it with both mouse click and finger tip on the screen ...
(Just to emphasise it: All this applies to Windows 8.1, I don't have this problem on my Windows 10 PC.)