Terminology: Name for a homomorphism from the free object?

This is almost identical to what a computer scientist might call a catamorphism. In this case, the algebraic structure is an initial F-algebra rather than a free commutative monoid. But just as you describe, operations like cardinality, sum and product of lists (as opposed to multisets) can all be described as catamorphisms. It's not an exact correspondence because commutative monoids don't form initial F-algebras in Set. (Though free monoids in general do, and I suspect that free commutative monoids form F-algebras in the category of monoids by Eckmann-Hilton.)

Mathematicians tend to call these kinds of things "canonical" maps.

The nLab has proposed the name "adjunct", which seems OK to me. So, for example, you could say something like "let $F(S)$ be the free group on the set $S$, let $G$ be a group, let $f:S\to G$ be a set map, and let $g:F(S)\to G$ denote the adjunct of $f$." I think it would be even a bit better to say "left adjunct" instead of just "adjunct".

I think Mac Lane might have used the term "transpose".