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.
I think you have to disambiguate two types of "breakage". One is suspending the thread, so to speak, and the other is functional destruction. I think perfectly non-destructive suspension of the process should not influence identity at all. Does an algorithm change its semantics if we pause the execution for a millisecond?
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.
If I duplicate the state of a paused algorithm and then immediately unpause both of them, even if I did it with no time in between, I can't claim that the second one is the first one. If the algorithm makes up some "mind", a new one is created - and there is a new person.
My point is basically that I will not magically look through someone else's eyes, even if the someone is my copy.
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.