I know that I can associate a popup.html to clicks on the extension button. However, I want it to be a menu, like the one popping up after clicking on Chrome's own "Customize and control Google Chrome" button (located to the right of your extension icons). I tried to Google it but no one seemed to be talking about that. Am I missing any common sense about Chrome extension development?
You can either register a click via
chrome.browserAction.onClicked but show no UI, or open an HTML page in a popup. You can style it to look like a menu, but it will still not be like a native drop-down menu.
Note that you can right-click the extension button,
but the menu you get there is fixed. I submitted a feature request a long time ago regarding that, but it never took off. There is a
"browser_action" now that achieves this.