The very act of being at place B instead of place A is gonna change you, regardless of whether you go there by transporter or by pony cart. So we should not concern ourselves with the tiniest details of sameness, as the events will obscure those from us (or from outside observers) anyway.
If the teleporter doesn't take your soul along for the ride, then it hasn't created a perfect copy, and supposedly this scenario deals with perfect copies only. So that's another thing we should not worry about. If the device neatly removes one body and creates another, but messes up some other details such as creating extra souls or losing the original, then it's a different question, and we should probably call a quantum mechanic to fix the machine before we try again.