I view it from a practical viewpoint: Even if you believe the Buddhist view, that the self is an illusion etc. you still feel like you have a self for >95% of the time (i.e. whenever you're not meditating). When you wake up in the morning you feel like you are the same person that went to sleep the evening before. On the other hand, a clone of you would not feel like it is you anymore than one identical twin feels it is the other. So ideally people in the future should create a person/simulation that feels like it went to sleep and woke up again when it "should" have died.

Problems arise mainly when you hit something that only partially feels like it is the same person. I'd say there is still a considerable range of possible people that are sufficiently similar that we say it is the same person, since there is also considerable variation in the normal functioning of human brains.

E.g.:

• Human memory is quite inaccurate. Different people with only slightly different memories could be said to be the same people. This may actually go quite far, if we consider the effects of Alzheimer's disease or other forms of amnesia.
• Being heavily intoxicated can to an extent feel like being a different person. Personality and habit changes over the course of your life can make you a different person, we still say it is the same person.

I wonder whether it is possible to find some sort of "core" personality/traits/memories, such that we can say as long as it remains unchanged it is the same person. I suspect there isn't, as it seems to be a gradient instead of a binary classification.