The question becomes more tractable if we assume that $f$ and $g$ are monic irreducible polynomials, and we restrict to $n$'s coprime to the discriminants of $f$ and $g$. In this case, by Hensel's lemma and standard facts on discriminants, the modified question remains unchanged if we restrict to primes $n$ coprime to the discriminants of $f$ and $g$.

Let us consider the corresponding number fields $K=\mathbb{Q}[x]/(f(x))$ and $L=\mathbb{Q}[x]/(g(x))$ whose discrimants divide the discriminants of $f$ and $g$, respectively. Then, in the special case when $L$ is Galois (i.e. $L$ contains all the roots of $g$), the modified condition holds for $f$ and $g$ if and only if $K$ contains $L$. This follows from Prop. 15 in Ch. VIII-5 of Weil: Basic number theory.

**Example.** Let $p$ be a prime congruent to $1$ mod $4$. Let $f(x)=x^{p-1}+\dots+1$ be the $p$-th cyclotomic polynomial, and let $g(x)=x^2-p$. Consider an arbitrary prime $n\nmid 2p$. Then, $f(x)$ has a root modulo $n$ if and only if $n$ is congruent to $1$ mod $p$, and $g(x)$ has a root modulo $n$ if and only if $n$ is a quadratic residue mod $p$. Indeed, in this case, $K=\mathbb{Q}(e^{2\pi i/p})$ is the $p$-th cyclotomic field and $L=\mathbb{Q}(\sqrt{p})$ is its unique quadratic subfield, and we just recorded the splitting primes in $K$ and $L$.