The Turing Machine is an abstract mathematical model for designing real machines. A Turing Machine is a design for a specific real machine. The computer or software of your choice is a real machine that is also a Turing Machine.
My toaster's special bread input tape is implemented using a mechanical lever. It's temperature oracle is implemented using a temperature sensor and a digital timer. I believe I have a toaster-enhanced TM.
The Turing Machine is really a machine, in every practical sense.
An apparatus using mechanical power and having several parts, each with a definite function and together performing a particular task.
So a mathematical notion definitely does not satisfy this definition. An abstract machine is definitely different from a machine. The map is not the territory (or was Korzybski wrong all along?).