We are experiencing the same problem with Chromium. When the drop down(select) is selected using a mouse, the window gains focus and recognizes a click, before briefly displaying an 'X' and failing to open the drop down. Very occasionally and without correlation to any event, the drop down will open. It seems to be related to the event handling in the current Chromium version. Works fine on late desktop versions of Chrome and Firefox.
Distribution: Raspbian Stretch
Browser: Chromium 60.0.3112.89
Server: Nginx 1.10.3
Hardware: Pi3 Model B V 1.2, Pi2 Model B V 1.2, Pi2 Model B V 1.1.