I don't know of any structure theorem, and I imagine any classification would be quite complicated. If you restrict your view to finitely generated cancellative monoids that are also saturated ($ka \in M$ for any $k>1, a \in M^{gp}$ implies $a \in M$) and sharp (the only unit is zero), then the question amounts to classifying finitely generated cones in $\mathbb{Z}^n$ up to $GL_n(\mathbb{Z})$-equivalence, and there doesn't seem to be an explicit answer to even this special case. Passing to the positive real or rational span removes a large amount of arithmetic information.
On the positive side, you have the fact that any finitely generated commutative monoid $M$ is finitely presented, i.e., there is a coequalizer diagram $P_1 \rightrightarrows P_0 \to M$ with $P_1$ and $P_0$ free commutative and finitely generated. The question of determining when two presentations yield isomorphic monoids is algorithmically decidable but seems rather difficult.