I think not.
FF doesn’t pass any information to the popup, so you can’t determine how it was opened.
browserAction.onClicked + browser.windows.create():
The browserAction.onClicked handler receives an OnClickData object.
But: Pressing the key combination and clicking the icon results in the same OnClickData.modifiers (none) and OnClickData.button (0).
If the user clicks the icon while pressing a keyboard modifier, you can detect this via OnClickData.modifiers
But that’s probably not what you want.
Also: In Chrome, browserAction.onClicked doesn’t receive an OnClickData object.
So you couldn’t use this method to make a cross-browser extension.
Why do you want to distinguish between the key combination and a click on the icon?