This question is motivated by problem 15.4.D(a) in Vakil, but to give some setup since the notation / terminology may differ: let $S_\bullet$ be a nice graded algebra (finitely generated, generated in degree 1) and $M_\bullet$ a graded $S_\bullet$-module.
The functor $\sim$ gives a natural way to turn $M_\bullet$ into a quasicoherent sheaf $\widetilde{M_\bullet}$ on $\operatorname{Proj} S_\bullet$ defined by $$ \Gamma(D(f), \widetilde{M_\bullet}) = ((M_\bullet)_f)_0 $$
(where the first subscript is localization and the latter is the 0th graded piece). This extends to the sheaves $ \widetilde{ M_\bullet (n) }$ via $$ \Gamma(D(f), \widetilde{M_\bullet (n)}) = ((M_\bullet)_f)_n $$
We obtain the saturation map $ M_\bullet \to \Gamma_\bullet(\widetilde{M_\bullet})$ where $$ \Gamma_n(\widetilde{M_\bullet}) = \Gamma(\operatorname{Proj} S_\bullet, \widetilde{M_\bullet (n)}) $$
This is indeed a (functorial) map of $S_\bullet$-modules since we always get a map $$ M_n \to (( M_\bullet(n) )_g)_0 \hspace{3em} x \mapsto \frac{x}{1} $$
for any $g \in S_+$, which induces the map $M_n \to \Gamma (\operatorname{Proj} S_\bullet, \widetilde{M_\bullet (n)} )$ . However, my main confusion is that Vakil says it is only in nice situations that $$ M_n \cong \Gamma ( \operatorname{Proj} S_\bullet, \widetilde{M_\bullet (n)} ) $$
so I'm not really sure how to interpret global sections in "not nice" scenarios. In particular, Vakil mentions that taking $S_\bullet = k[x]$ and either $M_\bullet = k[x]/ (x^2)$ or $M_\bullet = xk[x]$ we can show that the saturation map is neither injective nor surjective. Maybe there's just too much notation going on, but why is the map $$ k[x]/(x^2) \to (( k[x]/(x^2) )_x)_0 $$ not just inducing the usual $ x \mapsto \frac{x}{1} $ ?