I love firefox and use it daily on all platforms. What bothers me a bit (just a bit) about the design is that only on linux the menu bar row sticks out [1], while on Windows or Mac they were able to collapse it or hide it quite well. I suppose this is due to the GUI implementation constraints on the platform, but it is still very noticable when I switch from my mac to ubuntu.
I'm not on ubuntu right now, but this should be done by an ubuntu specific extension that places the menu bar in the global menu bar area that unity creates in essentially the same place it is on a mac. Should be installed by default on untuntu+unity, perhaps you disabled it?
I would guess that it's possible that the way X works prevents this from happening. At least, easily.
I use an offbeat windowing manager (WM) (aka, the client to Xorg that actually draws the base-level windows for the GUI). The WM gets to present titlebar as it pleases, and given how my WM works, there's 0 chance that Firefox would be able to draw it's own stuff up in there.
Given that, perhaps some WMs can allow the client window to draw up there and some can not. I get the impression that, at a minimum, the titlebar is not guaranteed to be fair game.
It's as B-Com said. The window manager. I used to use Emerald. There was no way you could consistently draw in the titlebar across all the themes available for it.
[1]: http://i.imgur.com/qDUuOXH.png