I don't think the polynomial given in the article was calculated via Remez. Perhaps the author merely meant it as an illustration. Here is Wolfram Alpha plotting the error of equation from the article:

https://www.wolframalpha.com/input?i=plot+sin%28x%29+-+P%28x....

You'll note that the error is small near zero, and large near pi/4, which is characteristic of a Taylor series around zero, and not the characteristic "level" oscillation characteristic of Remez approximations[1]. Note also that the polynomial only includes odd terms, which is not something I would expect from Remez unless it was run on the symmetric interval [-pi/4, pi/4].

I ran Remez for the problem and after 4 iterations obtained a degree 8 polynomial with error less than 1e-9, but it didn't look anything like the polynomial given in the article.

    1.7209863008943345e-05x^8 -0.00024575124459624625x^7 +7.194649190849227e-05x^6 +0.008268794893899754x^5 +3.425379759410762e-05x^4 -0.16667692317020713x^3 +1.5422400957890642e-06x^2 +0.9999999106213322x +8.526477071446453e-10

Although of course the first few digits of the low-order matching terms will be very similar - any polynomial approximation method will agree there because they are fitting the same function, after all. But by the time we reach x^5 or x^7 the agreement is very loose, really only the first couple digits are the same.

[1]: https://en.wikipedia.org/wiki/Approximation_theory