I agree that they'd both be you, but I don't think that's what's at stake. "You" would only get to experience one of the two lives at that point: either the original's or the robot's. Presumably whatever consciousness was present before the duplication still resides with the original.
So there's certainly an argument to be made that if someone created a copy of "you" and put it in a robot, and then destroyed your organic body, that consciousness wouldn't "move" to the robot; it would stay in the organic body and be killed. The robot would be a completely new consciousness, but with all of your memories and behaviors and attitudes.
I'm honestly not sure what I believe around this. Perhaps we will discover "consciousness" as a physical thing and learn how to transfer that as well. Or not. Who knows.
I'm with you on this one - continuity is important too, it's not just about form. A robot version of me might be a suitable substitute to my loved ones, but it would always be outside my head and therefore not a continuous extension of my self.
So there's certainly an argument to be made that if someone created a copy of "you" and put it in a robot, and then destroyed your organic body, that consciousness wouldn't "move" to the robot; it would stay in the organic body and be killed. The robot would be a completely new consciousness, but with all of your memories and behaviors and attitudes.
I'm honestly not sure what I believe around this. Perhaps we will discover "consciousness" as a physical thing and learn how to transfer that as well. Or not. Who knows.