To the best of my knowledge, we usually say that the expansion is most accurate around the expansion point (say, around $x=0$ in the case of the Maclaurin series).
I know the question is most likely stupid, but I often notice that the Taylor series is often used when working with infinite series where $n \to \infty$ and we are not in the neighborhood of the expansion point. What is the formal explanation to why we can safely do this without worrying about the accuracy etc. ?
We can only use a Taylor series to approximate a function within its "radius of convergence", which is the values of $x$ where the Taylor series actually approximates the function. So, we could only use the Taylor series at a "neighborhood of the expansion point", as you say, if it's within the radius of convergence.
This is why you can use the Taylor series for $\sin x$ and $e^x$ for any $x$ (since their radii of convergence include all of $\mathbb{R}$) but not for the series expansion of $\frac{1}{x^2+1}$, whose radius of convergence only includes $(-1,1)$.