At least in Faltings's setting, $m$ is the maximal ideal of a non-Noetherian valuation domain $V$. If we let $S$ be the non-zero elements of $m$, then the localization of $V$ at $S$ will be the field of fractions $K$ of $V$. The category of $K$-modules (i.e. $K$-vector spaces) is obtained as a Serre quotient of $V$-modules, but one quotients out modules which are killed by some non-zero element of $V$, while the category of amost modules is obtained by quotienting out by a much smaller category, namely the modules which are killed by every element of $m$.
In other words, I think you have misinterpreted the meaning of $m$-torsion, at least in so far it is used in the context of Faltings's "almost commutative algebra".
Added in response to the comment below: Let me not use the word $m$-torsion anymore, since it can be interpreted in different ways, and seems to be causing confusion.
Let $\mathcal M$ be the category of $V$-modules. Here are two Serre subcategories of $\mathcal M$:
$\mathcal C$, the category of almost zero modules, i.e. modules for which $m M = 0$. This is a Serre subcategory because $m^2 = m$ in the context of Faltings theory.
$\mathcal C'$, the category of torsion modules, i.e. modules such that $x M = 0$ for some non-zero $x \in m$. This is a Serre category just because $V$ is a domain.
Clearly $\mathcal C$ is contained in $\mathcal C'$, but they are far from equal!
The quotient $\mathcal M/\mathcal C$ is the category of almost modules. The quotient $\mathcal M/\mathcal C'$ is equivalent to the category of $K$-vector spaces (the equivalence being given by applying the functor $K\otimes\text{--}$, where $K$ is the fraction field of $V$).
In the statement of the question, you (mistakenly) define almost modules to be $\mathcal M/\mathcal C'$, and (correctly) conclude that this is equivalent to $K$-vector spaces. Once you replace this mistaken definition with the correct definition of almost modules (i.e. $\mathcal M/\mathcal C$), you will see that this is not equivalent to $K$-vector spaces.