I think of it this way:
1. The Order of Dagon presumably didn't have the godlike ability to create a human soul out of nothing; therefore, since Dawn was created and not born, she couldn't have had a soul in the normal sense.
2. The Key is very ancient, a ball of energy very much like souls are depicted in the Buffyverse, and is referred to by the monks as if it were a living being, not a thing.
3. If Dawn didn't have a soul, or something like it, she would be like a vampire, with no sense of morality or humanity.
4. So, assuming the monks can't create souls, The Key must be the thing serving in Dawn in place of a soul.
5. Since Dawn doesn't turn evil or amoral after "The Gift," The Key must still be there inside her, still serving as her soul. She doesn't think she opens anything anymore, but how would she know? She didn't know she was The Key in the first place until somebody told her. And Dark Willow obviously still thinks the power is inside her.
It's never been explicitly said (and I intend to ask if I ever get the chance), but it's possible that only natural-born human beings have souls, as such, in the Buffyverse. As is revealed in the preview pages for the upcoming issue of the season 9 comics, demons actually have their essences sent back to hell when their physical bodies are killed in this world (much like the demons in Supernatural when their "meat suits" get killed). Godlike beings, such as Glory and Illyria, probably don't have regular souls either, yet they are capable of making moral decisions.
Joss is an atheist, and I doubt he intends for the soul in the Buffyverse to necessarily be something "holy" like in Christianity. Instead it's like the rudder of a ship... the ship will still float without the rudder, but the captain won't be able to control which way it sails. Memories and the sense of identity clearly go with the body, not the soul... at least until the afterlife.