The structure theorem for finitely generated abelian groups furnishes for each $A$ an isomorphism $A\cong T(A)\oplus \frac{A}{T(A)}$ where $T$ is torsion. This is a family of pointwise isomorphisms between $1_{\mathsf{Ab}_\text{f.g}}$ and the functor $T\oplus \frac 1T$.
Claim. These functors are not naturally isomorphic. In particular, the isomorphisms of the structure theorem are not natural.
Proof. The endomorphism monoid of the identity functor is the multiplicative monoid $\mathbb Z$. This can be seen by looking at naturality squares mapping out of $\mathbb Z$ and using its universal property as the free abelian group on a single generator. On the other hand, the functor $T\oplus \frac 1T$ admits a nilpotent endomorphism $$T\oplus \frac 1T\overset{ \begin{pmatrix} 0 & \alpha \\ 0 & 0 \end{pmatrix} }{\longrightarrow}T\oplus \frac 1T$$ where $\alpha:\frac 1T\Rightarrow T$ is given componentwise by $\frac{A}{T(A)}\to T(A)\oplus \frac{A}{T(A)}\to T(A)$. Thus $1,T\oplus \frac 1T$ have non-isomorphic endomorphism monoids whence they are themselves non-isomorphic functors.