I wonder where this strong intuition about consciousness comes from.
Your "the current stream of consciousness" idea requires existence of something beyond all the information you have conscious access to and the procedures your brain performs. Namely, some "marker" which makes this particular realization of your consciousness unique, and which cannot be copied in any circumstances.
Well, that's the question. Since theoretically if I can be reconstituted perfectly, I can be copied perfectly, it is more incomprehensible to me to that two identical consciousnesses exist (two Me's with a capital M) than it is that Me is unique somehow, I work from the second assumption. Questions that still need answering include "what about slow cellular replacement with electronic parts?", to which I have no answer.
I think there's no marker. Both instances of you feel like each of them is you, if the copying was sufficiently faithful.
Concepts of the original and the copy are external to them. When you imagine the situation you take a place of external observer who has the information about who is the original and who is the copy.
ETA: It's something like imaging your death. You cannot truthfully imagine it first person, and (imperceptibly) slip into third person view.
Your "the current stream of consciousness" idea requires existence of something beyond all the information you have conscious access to and the procedures your brain performs. Namely, some "marker" which makes this particular realization of your consciousness unique, and which cannot be copied in any circumstances.
What do you think this "marker" is?