One thing that needs to be remembered when dealing with metaphysics in Kingdom Hearts is that the naming conventions are rather different. The heart is the thing that we generally consider the soul - it contains the things that make a person who they are, and it's the thing that would go on to any kind of afterlife. The soul, on the other hand, is more like what we'd consider the spirit - it's more of a life-force/breath of life thing than anything that represents the person (and as such, no one really cares as much about what happens to it as what happens to the heart).
As such, the Unbirths very much could have a connection to the soul. After all, "never received the breath of life" would be a rather fitting definition for "unborn." I'm not sure what, exactly, that would make them, though.