> My assumption is that we’d both feel like “me”.
You assume that both would feel like you, but there is no way you can prove it. The other can be a philosophical zombie [1] for all you know.
Would the "current you" feel any different after the duplication? Most people, including me, would find this counterintuitive. What happens if the other you travels to the other end of the world? What would you see? The question is not how the replica would think and act from an outside observer's perspective, but would it have the same consciousness as you. Would you call the replica "I"?
Or to make it more complex, what would happen if you save your current state to a hard disk, and an exact duplicate gets manufactured 100 years after you die, using the stored information?
[1] https://en.wikipedia.org/wiki/Philosophical_zombie