We aren't the fortress, because the we wouldn't exist until the miners dug the first room. Who, then, would be giving the orders to dig that room?
Instead, I have always though of us as an unnamed god, spirit, or essence. The embark represents our domain of sorts, with the dwarves as our subjects. We aren't very powerful gods, so rather than directly influencing the world we give strong suggestions to our subjects.
However, this also holds untrue because without player input the dwarves do not build their own fortress. After embark (the original seven having been motivated by an unknown call to the wild), the dwarves will sit around and starve without input. This suggests that as players, we replace dwarven motivation with our own, tearing out the old and implanting our own preconceptions and plans.
We aren't the fortress, we aren't simply the god, we're the Dwarven Mind as taken over by a God. Every mind is us, we are every mind. At once and forever.
This is good, because it also explains other strange occurrences during the fort. Strange moods aren't actually possessions from a fey spirit, but the original dwarven mind returning to control. The mind now back to it's original owner goes into a panick, and tries to make us (the god) as happy as possible so as to spare a brutal death. They become legendary only to stave off the bored and wandering mind of the indomitable overseer.
Vampire dwarfs are similar - their mind never returns, but their basic chemistry has changed to that of something else besides a dwarf. While you can control them most of the time, they reach a point of hunger where the vampire takes over and you (the god) loses control.
As you appoint nobles, you grant them a small amount of free will so as to take some of the strain away from yourself. As such, they start to make demands: surely they are necessary to the fortress, being given free will? They refuse to believe that you (the god) would actively seek their demise. And how wrong they are.
Dwarven traders? Why aren't they taken over as they step onto the embark? Easy - their minds are already full of compulsion planted by another such god.