Let me elaborate on unknown (google)'s comment. If $\text{Rng}$ denotes the category of unital rings and $\text{Grp}$ denotes the category of groups, then it turns out that "group of units" is a functor $\text{Rng} \to \text{Grp}$. In fact, it's representable; the group of units of a ring $R$ can be identified with $\text{Hom}(\mathbb{Z}[x, x^{-1}], R)$ (the set of all ring homomorphisms from $\mathbb{Z}[x, x^{-1}]$ to $R$), where the group structure on the Hom-set comes from a Hopf algebra structure on $\mathbb{Z}[x, x^{-1}]$. And covariant representable functors preserve limits, including binary products. (Actually, whether this argument is valid depends on the answer to this question which I posted yesterday.)
(A closely related functor is $\text{Hom}(\mathbb{Z}[x], R)$, which is none other than the forgetful functor $\text{Rng} \to \text{Set}$ which sends a ring to its set of elements. The fact that this functor preserves limits is precisely why the product of rings can be constructed using a product of the underlying sets. In more complicated categories where the obvious forgetful functor doesn't preserve limits, this isn't necessarily true.)
Closely related to representability (although I've never been clear on the precise relationship) is the fact that "group of units" has a left adjoint $\text{Grp} \to \text{Rng}$ which constructs the group ring of a group. And functors which have left adjoints preserve limits.
Presumably many other examples can be handled in a similar vein, although I can't think of any right now, so you'll have to give me more examples.