HERE IS THE FIX ...

I was experiencing exactly the same issue. 

When I was viewing an item for sale at Ebay.co.uk using Mozilla Firefox, and there was a drop-down menu choice e.g. 'size01 size02 size03'.  The menu would not respond by clicking with the cursor.

I am using Mozilla Firefox, and my browser window was not maximised, when I maximised it to the screen size, the drop-down menu functions normally, and everytime I came out of maximised, the drop-down menu did not work. 

The only website I experienced this with was Ebay.co.uk.

That is some pretty damn lousy coding for 2019!

SOLUTION:  Maximise your browser window.