The problem is that we're talking about a phenomenon that cannot even be measured: human consciousness as the conscious person himself experiences it.
The philosophical zombie is putatively a being that, though in all other respects identical to a human, including behaviorally and in terms of response, lacks true consciousness that is the voice in your head whispering to you or the thing that looks out through your eyes. However, the p-zombie writes books, partakes in philosophical discourse, and loudly claims that it is conscious the same way everyone else is. The problem is that no test will prove whether the p-zombie has or has not true consciousness; we can only prove that it acts in exactly the same way as conscious person.
There is the same problem with the teleporter. The clone that pops out the other side will, to everyone who interacts it, be the you who moved through the teleporter. It has your thoughts, your memories, your dreams and desires; in all measurable respects it is you. But nobody can prove that the thing inside its head, that whispers thoughts and gazes out through your eyes, is the same as that of the original you who entered the teleporter. That consciousness may have died, and a new one may be inhabiting the body of the teleporter-clone, and only you by virtue of your non-existence experience (or don't experience) the consequences.
The problem of the transfer of consciousness is totally different from that of the grandfather's axe. The latter is a paradox of categorization. The former is a true mystery.