According to wiki, given $M$ a commutative monoid, the Grothendieck group $K$ is an abelian group satisfying a homomorphism, $i:M\rightarrow K$ s.t. for any homomorphism $f:M \rightarrow A$ into an abelina group $A$, there exists a unique homomorphism $g:K \rightarrow A$ such that $g \circ i = f $.
Sorry, maybe this is obvious, but why couldn't we take $K = Z(M)$, the free abelian group generated by $M$, instead of quotienting out a subgroup? Does not the universal property of $K$ follow from the universal property of $Z(M)$?
The canonical inclusion $i:M\to Z(M)$ is not a homomorphism. Given $a,b\in M$, $i(a)+i(b)$ will be the formal sum of $a$ and $b$ as elements of $Z(M)$, which is different from $i(a+b)$. Basically, $Z(M)$ completely forgets about the monoid structure of $M$ and treats $M$ as just a set, and so $i$ will not preserve that structure.