Edited since you updated the question which addressed my answer.
I did some research. It looks like they've been playing with the system tray icon functionality since before version 17. (https://code.google.com/p/chromium/issues/detail?id=72068, https://code.google.com/p/chromium/issues/detail?id=142450). Even then they had the issue with the icons being tied to the chrome icon. I think the reason is that the extension runs through chrome.
There's obviously a way around this since the chrome icon can go away and leave the chrome icon if the chrome task manager icon goes away when you disable background tasks.
You could use the native app:
(It doesn't seem like it has the same functionality and for some reason can't connect to the service even though I can get there through the browser, but it doesn't have the same systray issues with the chrome task manager icon)
And/or submit a bug report to google:
There are other third party solutions like the minimize to tray solution which doesn't seem to do exactly what you want it to, but it's possible to probably throw something together from that code to make it happen.
In windows 7 and 8, click the up arrow and go to customize, find the chrome icon and change the setting to never show.
In earlier versions of windows, right click task bar, properties, customize. You can tell various icons to never show there.