There are already several answers describing how ultraproducts arise naturally from categorical considerations. However, I don't think any of them address this part of the question:
> I would be very interested, if there were a natural universal characterization in terms of the usual Hom sets

In fact, there is a universal characterisation of ultraproducts in any category, which looks similar to the characterisation of the universal properties of colimits in terms of universal cones. This is described in §5 of Tendas's [Flatness, weakly lex colimits, and free exact completions](https://arxiv.org/abs/2304.09000).

> **Definition 5.1 (Tendas).** Let $\mathcal M$ be a category and $\mathcal U$ be an ultrafilter on a set $X$. Given an $X$-indexed family of objects $(M_x)_{x\in X}$, the *universal ultraproduct* of $(M_x)_{x\in X}$ over $\mathcal U$ is an object $\textstyle\prod_{\mathcal U} M_x \in \mathcal M$ together with a cocone $$(\delta_S \colon \prod_{s \in S}\mathcal M(-, M_s) \to \mathcal M(-, \textstyle\prod_{\mathcal U} M_x))_{S \in \mathcal U}$$ over $\mathcal U$ which is universal among all other cocones with representable codomain.

When $\mathcal M$ has products, universal ultraproducts coincide with the *categorial ultraproducts* of Lurie's [Ultracategories](https://people.math.harvard.edu/~lurie/papers/Conceptual.pdf) (Proposition 5.4 of Tendas).