I'd have to say no, considering souls are a myth anyway. NO ONE has a soul.
that's a fish, not a soul
Yes - at the moment that it recognized that it had been teleported and began making seemingly independent decisions. It's soul could not then be the same soul in the context to which you refer because its life experience would be different (as would the original copy whose life would have different experiences.
Think of this experiment already done. A photon is split and, using mirrors, is directed along two different courses inside a box. As the (seemingly) two photons are moving around, each has a different probable future and probable past. Join them up at the end again and you again have one photon that has three pasts. (One photon's, the other photon's, and the spooky-at-a-distance observer who was always present (through entanglement) though unseen by us 2 or 3 dimensional creatures.
This is very much in-line with the split-universe theory. When a decision is made, the universe splits and copies go forth in their own directions depending on their expectations. Neither can see the other, yet they are still entangled.
1+1=at least 3
Who knows how it would work for living things. There is a very large difference between moving photons, atoms... and an entire human body and all it's cells. We do not even understand how the brain really works and if we have a "soul" or if it is just electric connections.
we are just brains in bags of water stop being all high mighty about how super important the human race is. So we got lucky by pure dumb natural selection...now suddenly we have souls and purpose in life beyond simple survival? Surely you can see the arrogance in the human train of thought? This is defeat by victory....the irony!
We're just complicated Lego with wires! Put em back together and your just that. If you couldn't do that, you wouldn't be you but multiple things!