I actually think it's pretty simple.
Consciencesness = software.
During the teleportation, the body and mind exist in both places. Ergo, continuity (statefulness) is maintained, and the single consciencesness exists in both places simultaneously. Think VM on a live migration. As long as the two copies are forced into exactly the same state (entangled?) It is really just one linked mind.
My opinion is that the thing we call a soul, or self awareness, is something that lives entirely in the software of the mind; it's not a tethered spark of ether in an intangible universe. Understanding that our souls are really just the software of an electrochemical network doesn't deaden the experience of self awareness; accepting an understanding of how the mind works shouldn't make your self awareness any less poignant.
I do think, however, that the software is less continuous than we may want to admit. Sleeping may be analagous to a computer low power states, but accepting that the soul is effectively a form of software running on a form of a network also accepts that it is likely there are instances where it can be wiped, altered, rebooted, or replaced.
There is a lot of interesting reading that dances around this premise, such as, "The Man Who Mistook His Wife For a Hat." I think it is likely that if we withstand significant trauma or injury the person that wakes up is potentially an altered iteration.
However, studies of injuries and neurological issues suggests that there are many levels of how we are encoded in the brain, some of which persist on a durable, low level. Stretching the computer analogy to it's breaking point, short term memory is kind of "Cache", while mid term memory is "RAM". Long term memory is some kind of storage, while behavior, personality and manner are probably closer to System Code. One wonders if the analogy fits well because we build computers to match the ways we think.
If the mind works like this, there is a big upside: it makes the path to uploading of the mind straight forward. All we need is the capability to completely simulate a human brain, and to synchronize the two versions completely. One would experience both sets of inputs simultaneously, and the self would be in both places simultaneously. Gracefully shutdown the meat version (cryopreserve?) and you will have moved the mind/soul. Details of the technical implementation are left as an exercise for the reader.