You might need to give a little more information. What version of Ubuntu? 18.04? What desktop? Unity? Gnome? Budgie? KDE? I’m using Ubuntu 18.04 with Budgie desktop and using Plank as the ‘dock’. I assume this is what you mean by toolbar (everybody calls it something different). The Thunderbird icon does show a number and it does seem to update in real time. Budgie panel, on the other hand, doesn’t support this at all — no number at all. The problem is probably with a particular desktop not supporting those dynamic updates, so running Thunderbird as a daemon would make no difference. It would also be a bad idea as it’s not designed for that.