Nope - the Taylor series depends on a specific choice of point. Around that point, the truncated Taylor series is the "best" approximation of the function using a polynomial.

In this case, the author has picked seven points on the sine curve (or its integral or its derivative), and used those seven pieces of information to find a unique polynomial.

If I were to guess, I'd say that if the seven points were on the curve itself (not on its integral or derivative), and they were all very close to some point P, then the resulting polynomial would look a lot like the taylor series expansion around P.

But that's just a guess :)