To answer the question raised in the most recent edit: abelian sheaves are tensored and powered over abelian groups.

First of all, abelian presheaves have tensors and powers that are computed *pointwise*. By "abelian presheaves" I mean the $\textbf{Ab}$-category of functors $F: C \to \textbf{Ab}$ where $C$ is a small (ordinary) category, or somewhat more generally the $\textbf{Ab}$-category of $\textbf{Ab}$-functors $F: C \to \textbf{Ab}$ where $C$ is small and $\textbf{Ab}$-enriched (we can always regard a small $\textbf{Set}$-category $C$ as a small $\textbf{Ab}$-category, by defining $C(a, b) = F\hom(a, b)$, the free abelian group generated by the set $\hom(a, b)$). If $A$ is an abelian group, then the power $F^A$ is given by the "pointwise" formula $F^A(c) = F(c)^A$, i.e., $F^A(c) = \textbf{Ab}(A, F(c))$. This is a straightforward calculation having to do with commutation of limits. We have for each $G: C \to \textbf{Ab}$ an isomorphism $\textbf{Ab}^C(G, F^A) \cong \textbf{Ab}(A, \textbf{Ab}^C(G, F))$ since

$$\begin{array}{rcl}
\textbf{Ab}^C(G, F^A) & \cong & \int_c \textbf{Ab}(Gc, F^A(c)) \\
& \cong & \int_c \textbf{Ab}(Gc, \textbf{Ab}(A, Fc)) \\
& \cong & \int_c \textbf{Ab}(Gc \otimes A, Fc) \\
& \cong & \int_c \textbf{Ab}(A, \textbf{Ab}(Gc, Fc)) \\
& \cong & \textbf{Ab}(A, \int_c \textbf{Ab}(Gc, Fc) \\
& \cong & \textbf{Ab}(A, \textbf{Ab}^C(G, F)).
\end{array}$$

A very similar sort of calculation shows that $\textbf{Ab}$-tensors of abelian sheaves are also computed in pointwise (or objectwise) fashion.

As for sheaves: if $F$ is an abelian sheaf (for whatever topology), then for any abelian group $A$ the presheaf $F^A$ is a sheaf. Abstractly, if $F$ is an algebra of the (enriched) sheafification monad $\sigma$ on $\textbf{Ab}^C$, then so is $F^A$. For we have a universal element $\mathbb{Z} \to \textbf{Ab}(A, \textbf{Ab}^C(F^A, F))$ and a map

$$\textbf{Ab}^C(F^A, F) \to \textbf{Ab}^C(\sigma(F^A), \sigma(F)).$$

Putting these together we get $\mathbb{Z} \to \textbf{Ab}(A, \textbf{Ab}^C(\sigma(F^A), \sigma(F)) \cong \textbf{Ab}^C(\sigma(F^A), \sigma(F)^A)$, i.e., a map $\sigma(F^A) \to \sigma(F)^A$. Together with the algebra structure $\alpha: \sigma(F) \to F$, we produce an algebra structure on $F^A$ by an evident composite

$$\sigma(F^A) \to \sigma(F)^A \stackrel{\alpha^A}{\to} F^A$$

where $\alpha^A$ for any $\alpha: G \to F$ is defined by exploiting maps

$$\mathbb{Z} \stackrel{\text{canon}}{\to} \textbf{Ab}(A, \textbf{Ab}^C(G^A, G)) \stackrel{\textbf{Ab}(1, \textbf{Ab}^C(1, \alpha))}{\to} \textbf{Ab}(A, \textbf{Ab}^C(G^A, F)) \cong \textbf{Ab}^C(G^A, F^A)$$

(cf. similar constructions mentioned at Morphisms of cotensors).

Nothing in this answer so far has much to do with the details of abelian groups or sheafification or anything like that; it's really just pure enriched category theory. A more general result is that if $\mathcal{V}$ is a suitable base of enrichment (complete, cocomplete, symmetric monoidal closed) and if $C$ is a small $\mathcal{V}$-category, then $\mathcal{V}^C$ has $\mathcal{V}$-powers and $\mathcal{V}$-tensors (in fact is $\mathcal{V}$- complete and cocomplete), and also if $\mathcal{C}$ is $\mathcal{V}$-complete and $\sigma$ is a $\mathcal{V}$-monad on $\mathcal{C}$, then the $\mathcal{V}$-category of algebras $\mathcal{C}^\sigma$ inherits $\mathcal{V}$-weighted limits from $\mathcal{C}$. Just as you would expect by analogy from ordinary category theory. Putting these results together covers a huge range of examples that arise in the wild.

As for tensoring on abelian sheaves: this is a special case of (weighted) colimits on sheaves, where the recipe is well-known: sheafify the (pointwise) colimit taken in presheaves. Thus the formula for a sheaf $F$ should be given by $(A \cdot F)(c) = a(A \otimes iF(c))$, where $i$ is the full inclusion of sheaves into presheaves and $a$ is its left adjoint, reflecting presheaves back into sheaves. This construction applies more generally to *idempotent* monads $\sigma = ai$ where $i$ is a full inclusion and $a \dashv i$; however, it doesn't work for general (enriched) monads $\sigma$ (and in fact the construction of colimits in categories of algebras is a pretty big topic in its own right). But anyway, if $G$ is a sheaf, then we have

$$\text{Sh}(A \cdot F, G) \cong \text{Sh}(a(A \otimes iF), G) \cong \textbf{Ab}^C(A \otimes iF, iG) \cong \textbf{Ab}(A, \textbf{Ab}^C(iF, iG)) \cong \textbf{Ab}(A, \text{Sh}(F, G))$$

where the last isomorphism invokes (enriched) full faithfulness.

Although much of the theory of weighted limits and colimits in enriched category theory is a smooth generalization of the theory of ordinary (conical) limits and colimits from ordinary category theory, it must be said that powers and tensors add an important enriched ingredient, in the sense that

Categories that are complete and cocomplete in the ordinary sense need not admit (enriched) powers and tensors, but

If a category is complete in the ordinary sense and admits enriched powers, then it is complete in the enriched sense of admitting all weighted limits. Similarly if a category is cocomplete in the ordinary sense and admits tensors, then it is cocomplete in the enriched sense of admitting all weighted colimits.

For an example of the first point: consider the inclusion of monoids $M$ into categories, taking $M$ to the usual one-object category $BM$ whose morphisms are elements of $M$. Monoids are thus $\textbf{Cat}$-enriched by the formula $\textbf{Cat}(BM, BN)$. Certainly $\textbf{Mon}$ admits all ordinary limits (they are computed as in $\textbf{Set}$). However, $\textbf{Mon}$ does not admit $\textbf{Cat}$-powers. For instance if $\mathbf{2}$ is the arrow category, then the objects of $(BM)^\mathbf{2}$ correspond to elements $m \in M$, hence have more than one object (and in fact need not be even *equivalent* as categories to monoids, when one considers that morphisms $a: m \to n$ in $(BM)^\mathbf{2}$ are elements $a \in M$ such that $am = na$). This is enough to show $\textbf{Mon}$ doesn't have copowers, because the full inclusion $B: \textbf{Mon} \to \textbf{Cat}$ reflects any weighted limit in $\textbf{Cat}$.

Essentially all of this material can be found in Kelly's book, Basic Concepts of Enriched Category Theory.