Regarding the specific example, the construction of $C'$ can be tightened up as follows. The
first functor, from $C$ to $C'$ is not just hom-preserving (closed) but bijective on objects. The second functor, from $C'$ to $D$, has the property that it is fully faithful on maps out of the monoidal unit.

(This is enough to determine $C'$ uniquely. As is implicit in Scott's comment, you need more than just the fact that the first leg is closed, otherwise you could take the first map to be the identity.)

So you could consider the (2-)category of symmetric monoidal closed categories, and (strong) symmetric monoidal functors. Any such morphism $F:C\to D$ comes with a canonical comparison
$F[c,d]\to[Fc,Fd]$, and we can call it (strongly) closed if these comparisons are invertible. There's a class E of morphisms consisting of those which are bijective on objects and closed, and a class M of morphisms consisting of those for which $F$ induces a bijection between maps $i\to c$ and maps $Fi\to Fc$, for all $c$, where $i$ is the monoidal unit. (You might call these maps pre-fully-faithful, or something like that.) These classes E and M are closed under
composition. I haven't checked in detail, but it looks like they have a reasonable chance of
being orthogonal to each other, and so you would have a factorization system.

I don't see a way of doing something similar with mere functors between monoidal categories. But perhaps this is too much to expect. In the other example, although the internal hom is not preserved, there is a canonical comparison. So perhaps rather than looking at plain functors you should look at the lax monoidal ones (often just called monoidal, with the word strong being added to mean preservation up to isomorphism). Now it is true that every lax monoidal functor $C\to D$ factorizes as a lax monoidal $C\to C'$ followed by a strict monoidal $C'\to D$, moreover in a universal (initial) way. For any monoidal category $C$ there is a lax monoidal $p:C\to C'$ with the property that composition with $p$ induces a bijection between lax monoidal $C\to D$ and strict monoidal $C'\to D$, for any $D$. (Notice
that the order is opposite to that in your example: the strict map comes second.)

This situation is quite common, and holds for many different types of structure. It is much
less common, but does sometimes happen that there's a universal factorization $C\to D'\to D$ with the map $C\to D'$ strict and the map $D'\to D$ non-strict.

Your example involves an extra ingredient involving the bijective-on-objects and pre-fully-faithful conditions.