I can't really contribute to this thread much, because I hold the very extreme opinion that continuity and even memories aren't necessary for a person to be itself, but their base goals and thought patterns are.
Lets say you have a person called Bob. Bob gets hit in head with a brick and loses his memories. He still likes chicken and long walks on the beach. If given a choice of what to eat, he would take chicken, exactly like Bob. He simply does not know his own name.
Now lets say Bob didn't lose his memories, but instead became a vegetarian because he could no longer stand the taste of meat, and now has a phobia of sand. He shares the exact same memories with the original Bob, but acts completely differently, and therefore I consider it another person.
Now lets take it a step further. Say across the world another person very similar to Bob, Carl, who just has different experiences, gets hit in the head with a brick, and now acts the same as amnesiac Bob. Since if A=B and B=C therefore A=C as in, since amnesiac Bob is Bob and Carl is amnesiac Bob, therefore Carl is Bob. This same principle would apply to clones/copies/virtualized brains. If similar inputs give similar results, then its the same person.
Obviously this does not apply if you don't think a person and its amnesiac self are the same, but unless you go with that, then a person is actually many people, due to forgetting their past experiences and gaining new sets of memories.
And of course, if there's two of a person running around I wouldnt consider the the exact same, much as you wouldnt call two identical chairs a single chair, but a part of larger set, in the case of my examples there is a set "Bob", which contains Bob, amnesiac Bob and Carl, but not brain damaged Bob. It could also contain computers that think like Bob or sentient slimes. It also contains the copies of Bob that the discussion is about, especially if the memories are the same.
Just my 2 cents. Right now everyone seems to have a different concept of what constitutes "the same person", and I don't really think this is a thing that can be really debated away. But I think its fun to take peoples thoughts to their logical conclusions(like mine, where a sentient cloud of dust could be Bob).