The process of creating these machines is the crucial part because if a singular process can create infinitely many machines, then those machines must bear some sort of commonality. This is what computer science is hoping to study, but we constantly get distracted by studying details of individual machines, individual numbers. And often, we confuse ourselves into thinking we haven't found any commonality at all -- that our machines _aren't_ Turing machines! But of course, this is incorrect.
When Turing was creating the machine to counter the Enigma Machine, do you think he just thought of it as a "computer"? or did he think of it as a machine created by his process, a Turing Machine? (though he probably wouldn't have used that term)
Edit: a concrete refutation of your claim that computers are not Turing Machines would then be to create a machine that solves the Halting Problem. Turing Machines cannot solve the Halting Problem so if our computers are not Turing Machines then they must be able.
We arrived at the Halting Problem because of our study of Turing Machines as a class of machines and our computers are within that class.