To say somehow the different history of the atoms in the original brain vs the atoms in duplicate is somehow relevant to the emergent property would be a very odd claim: we don't take into account the prior history of the atoms in turbulent flow for example, just what the are doing right now.
Or for clarity, if I gradually replaced every molecule in your brain with another identical one, the emergent "you" would not be affected. So, if it weren't already obvious, it clearly can not be dependent on the particular atoms that make you up.
And by the way, if your brain at 10 years old and 90 years old, in spite of being very different structurally, still give rise to your same "you", the copy would not even have to be very good.
It is a fun problem I must say.