Given that (methinks) consciousness is just a sensation generated by a feedback loop, with the exception of a mild confusion caused by the fact that discrete identity (Theseus' Ship) is just a human construct (as opposed to a continuum with everything connected), there is nothing special about this.
For the person who believes they're Jesus, they are really Jesus. How would you know that you are yourself? What if you suffer form amnesia? Make it temporal and you are really onto some mind-bending territory.
Teleportation suffers from the same affliction: https://existentialcomics.com/comic/1
Here's my take: Consciousness is real only to the one experiencing it. Just like Love - yeah, it's as real as anything can be for the subject experiencing it, but for everybody else the only real thing about that Love is how it manifests.
Maybe you can separate it from the 'machine' running it, put it in another machine, but is it the same? Yes, of course, for the machine(s) running it, not so much for the outside observers that cannot experience a given sensation outside themselves.
The individual =/= its (current) consciousness, therefore uploading consciousness achieves just as much as 'uploading' my genes into a new individual (a child). Something will go on existing, but it's not me. It's another machine that might have my memories and SENSE of identity.