Not everything but some things. Think about this for a minute: if there's no way to tell one object from another, then we must think of them as the same object. That is known in physics as the Pauli exclusion principle, and that's where you seem to be coming from.
I would agree with you that if an outside observer can't tell the difference between myself and a copy of myself, then the outside observer must think of both me's as the same person. But the problem for your argument is that it doesn't take into account the subjective perspective: even if you think there are two me's, *I* am convinced there is only one. If you see my copy walking down the street and you tell me, "hey, look, that's you down there", I would immediately correct you, "no, that's not me, I am here by your side; what you see there is my copy".
It's not meaningless. I can take a piece of paper and draw a circle of radius X. I can take another piece of paper and copy my first drawing: draw another circle of radius X. Have I made a copy? I sure have. Are both drawings the same? They are not! Even if they look exactly the same, down to the last particle in the piece of paper, they still occupy two different positions in space and time.
From my perspective, the flaw in this "copying a person" scenario is the failure to take into account that you can copy a person's body but you cannot copy the space and time coordinates a person occupies.
I object to the "sharing close to the same properties". You can, at least in principle, copy any object down to every single property except for two: its position in space and time. And since one property of consciousness is that it knows its position in space and time, that means you can't copy a person's consciousness, even if you can copy every single atom of that person's body.
I'm sorry if that argument smacks of souls, spirits, or dualism. I can't do anything about it, it's pure logic as far as I can tell.