> How does such a incontrovertible structure arise?

yes, I also enjoy trying to answer this question.

what is such an structure even mean? how could it be that simply defining numbers, obersving addition, and generalizing it away into multiplication would yield this natural structure?

It all begins with zero. the predecessor of One, the best known number.

zero can be assumed by anyone. the surprise is how all zeros are the same zero. (by uniqueness of emptyset; but as I hope you can see, I'm a crank. a nutjob. I'll stop