This just doesn't fit the theme of Kingdom Hearts. I think Kingdom Hearts is a game more about the what happens to a heart when it leaves the body. Thus their intent was to leave the explanation of what happens to a soul when it leaves the body up to local religion of each individual world.
So far all that has been explained is how the heart would separate from the body (being engulfed in darkness), what happens to the heart after leaving the body (becoming a heartless), and what happens the the leftovers, the body and soul (creates a nobody).
Now a soul leaves the body through destruction of the body. However, what happens if the soul was forcibly removed from the body without going through death? This is where I think unbirths might come from. Rather than unbirths purely being the soul, they are the heart and body, both still in working condition. (for those people who are Harry Potter geeks, think the dementor's kiss)
This would fit Kingdom Hearts as it deals with yet another situation the heart could be placed in. Also would help explain the name unbirth. As a creature that has a heart and body, but no soul, has not been truely born yet.
the other, and less likely opinion is they are the heart and soul, without the body. But a soul purely on its own would just go to the "afterlife". And that's ground I don't think game developers want to step foot on.