This is different from $C$ being dualizable ($[C,D] = C^\vee \otimes D$). (EDIT: It turns out to be the same -- see Mike Shulman's answer!) But for example, if $C$ is a locally free sheaf of finite rank on a scheme/locally ringed space $X$, then $C$ has this property in the category of quasicoherent sheaves, or in the category of $\mathcal O_X$-modules -- just check locally.
Here I'm working in a symmetric monoidal closed category $\mathcal C$ with monoidal product $\otimes$, unit $I$, internal hom $[-,-]$, and $E^\vee$ denotes the dual $E^\vee = [E,I]$.
I'm really tempted to call such an object $C$ "locally free (of finite rank)", because the condition says that you can understand maps $D \to C$ as long as you internally (i.e. locally) understand maps $D \to I$, i.e. maps into the canonical "free" object.
Perhaps I should say "locally Cauchy-free" instead of "locally free (of finite rank)", since presumably the significance of "locally being able to take finite sums of $I$" is that finite sums of $I$ are $\mathcal C$-enriched absolute colimits (a.k.a. $\mathcal C$-enriched Cauchy colimits) in the additive context. But I don't understand what's going on well enough to firmly draw this connection.