Richtofen's eternal soul was collected from Dimension 2210 between Origins and The Giant, so by all accounts, all Richtofens should be soulless if we are following the rules. All this leads me to believe there are two, maybe three components to make a living being:
1. The Eternal Soul: Only seems to be collectible from one particular version at any time, and once it is collected, memories from all other versions start to bleed through to each other. As long as the Eternal Soul is present in the House, Dr. Monty can wipe away all other versions of that being.
2. The Soul/Aura: What makes each being special. While Richtofen can be soulless, his eternal soul stored in the house, there is still a special "Aura" present that makes him unique, and can be transferred between bodies. Richtofen has gone from his own body, to Samantha, to several Zombie bodies, and back to his own body. The Aura may actually look, outside the body, like the ghosts of Alcatraz.
3. Life Force: This may potentially be the same as the Soul/Aura, because for example, Zombies in Die Rise seem to be screaming out in terror, as if some part of their original "Aura" is there. However, I also point to Richtofen's empty body, which Pernell notes has "no brain activity", but is still very much alive. There is no eternal soul or aura present, but something is keeping it alive. This may also be what is collected in canisters, and what is given to ADAMs by the Elemental Shard.