The crux of the issue for me is that the two can exist at the same time. If your copy is "alive" and you are alive as well and a separate entity, then how can you say that it is you?
OK, but like with the lego house - what if you just swap pieces out one at a time? What if you gradually upload parts of your brain onto the computer, and those parts cease functioning in your biological body?
Also, your body is constantly swapping out parts. You aren't made of the same cells you were made up of 10 years ago. So what kept you, you?
The other thing about making copies is that the "essence" difference is not actually important. If you have a copy of your lego house, sure it's not "the same" one. But for all practical intents and purposes, it might as well be the same house. It's just an artifact of human thinking that insists there is a fundamental essence of each object, that it is not just the sum of its parts. That's why people insist on believing in souls and such.
But if replacing all the cells in your body doesn't cause you to stop being you, then what if you were duplicated, creating an exact double with all of your memories - all the way up to the moment of duplication, such that both of you had full memories of your whole life and of being cloned? Then which one is you? Given that you two are completely identical, if one were then snuffed out, how would it make any practical difference, since whichever one lived would, in its own mind, be the real one, since that person would have a full life of memories? And if that happened, how would that person be "you" any less than you are now, having replaced all the parts you're made up of, many times over?