The exact copy and same past arguments are red herrings, the new person is still a new person.
Suppose that the copy could be exact (i.e. the duplication process is perfect) - on this basis you provided the argument above. Now suppose that it is not perfect - maybe something trivial like transportees get blue hair. For your argument - does this matter and if so, how / why? If the blue hair is an acceptable compromise,what further compromises are allowable? At what point does the copy stop being 'you'?
I don't know exactly, and who would? But if someone kidnapped you and dyed your hair blue against your will, are you a different person?
But besides that, if the duplicator doesn't work,
it doesn't work. This discussion doesn't mean anything. The point is if it were perfect (and it conceivably
could be), how does that affect your sense of self?
Before my conception I share the same past as my Father. In your example, the copy does not share the same physical past as the original. It's physical past is the engineering of the duplication machine and the use of it to manufacture the copy. True, the copy is made with a memory that is a copy of the original's memory but it is not the same memory.
But your father's consciousness is incredibly different than your consciousness. It's like saying "this apple is different than that orange". You'd never dream of claiming that you and your father have the same consciousness.
For what philosophers call monists (not duelists), the only thing that defines a consciousness is the information of a person: their memories, habits, knowledge, and so forth. You don't share
any of that with your father. What you share with your father is DNA, which is unrelated. For the perfect duplicate, you
do share all this information. If you want to define a person in the way that people would recognize, the copy and the original are have the same consciousness.
You can imagine all sorts of fantastical situations (teleporters, time travel, wormholes in the space-time continuum, etc) where the fine distinctions you are drawing wouldn't apply (time travellers don't have a consciousness which is continuous in time, for example), but no one would ever say "well that's just a doppelganger, not the original".
If the original is destroyed then what is achieved is not really transportation but an illusion of transportation that an outside observer (but not the 'transportee', for he is dead) can experience. If the original is not destroyed then the illusion fails and you have simply created a remote copy of yourself. In the Prestige the duplication machine is used precisely for this purpose - to create an illusion of transportation.
I haven't seen the prestige, but my point is while this sort of teleportation would only really be a trick,
as far as a person, and not their body is concerned the difference doesn't matter. To the person before the transportation, they will wake up somewhere else.