The point is that we can lose our continuity of consciousness, yet we don't consider this "dying" in normal life. If we duplicate our exact physical brain and processes and immediately destroy the original while this happens, there is no loss of continuity, but this feels like dying still.
To me, there is clear loss of continuity when you duplicate a brain. The process inside has been broken/does not continue. The structure is not the key, the key is the running state of the brain. Under anesthesia, the brain still functions.
This also assumes that duplication requires either type of breakage--for thought experiment purposes you may assume that cloning is possible with 0 delay and with 0 destructive change.
If you care about which physical substrate the process is running on specifically, this is the bias towards physical continuity I noted.
My point is basically that I will not magically look through someone else's eyes, even if the someone is my copy.