It's more interesting to consider the case of making two cloned copies. They each start with the same exact state, identical to you; but from that moment on, they're independent. They each have an I.
Now, so do you. If you're then killed, you're dead. Two new I's come into existence; each initially feeling and thinking exactly as you at the moment of cloning; and for every moment thereafter, a separate human being. An I. A subjective consciousness.
As it happens we have a real world model in common use. In computer programming, a process is a thread of control with its own private memory space. Typically a process is a running program, such as a copy of your browser or an instance of a word processing program
Now, a process can create another subprocess. (In Unix-like systems this is called a fork, but other operating systems have analagous constructs.)
At the moment of forking the subprocess inherits the entire memory space of the parent process. At that moment, these are two separate processes, or running programs, executing in that machine. They have the exact same state at that moment. But from the moment of cloning and onward, these are two separate processes. They are totally independent of each other (except for the relationship of "is the parent of" that the OS remembers).
No programmer would ever think they're "the same process." They're two separate processes ... independent software entities running under the operating system.
They do happen to
a) Run the exact same code. They execute the same program. They have the same code.
b) The subprocess is initialized to have the exact same state as the parent process at the moment of cloning.
It seems to me that the field of computer programming already has a rich metaphor for the notion of cloning.
A cloned human is a new human that runs the same code (DNA and basic brain wiring) and initially has its state set equal to the state of the parent human. From that moment on, the two humans function independently, have different life experiences, develop their separate minds independently, act independently in the world. Exactly like a parent and child process in a Unix-like operating system.
And for that matter, exactly the way a human makes a new child now! You spawn a new human and initialize it to a new-born infant state. Everyone instinctively understand that the baby is a new human being. Likewise if you cloned yourself, your clone would be at that moment a brand new human being; just one whose initial state was set equal to your current state. But it's a different I. And if they kill you, you're dead. It's your clone who will live.
This is conceptually no different than a human giving birth. It's merely a question of setting the initial state. [For that matter, why not clone me but make me a lot younger!! There would be a market for that :-)]
Now, how strong is my analogy between operating system process cloning and human mind-cloning or Star Trek transporter technology?
As a metaphor it's solid. I think it provides great clarity when thinking about what it would mean to transfer a human consciousness into a different substrate.
Whether it's literally true in some way ... we have no way of knowing. The science of consciousness (the "hard problem," as they say) is a tough nut to crack. We shouldn't be deluded by the fact that we have cool computer technology, into thinking that we'll soon be uploading minds.
We don't even know what a mind is.