In fact, it does and not only for Hopf algebras. Let $(B,m,u,\Delta,\varepsilon)$ be a bialgebra over a field $\Bbbk$ (for the sake of simplicity, I'm not sure it is really needed).
Lemma: For every left $B$-module $V$ the functor ${_B\mathsf{Hom}}(B\otimes V,-)$ is right adjoint to the functor $-\otimes V:{_{B}\mathsf{Mod}}\to {_{B}\mathsf{Mod}}$. The left $B$-action on $B\otimes V$ is the diagonal one and the left $B$-action on ${_B\mathsf{Hom}}(B\otimes V,W)$ is given by
$$(b\cdot f)(a\otimes v) = f(ab\otimes v).$$
Proof: Define the assignments
$$\varphi:{_B\mathsf{Hom}}(U\otimes V,W)\to {_B\mathsf{Hom}}(U,{_B\mathsf{Hom}}(B\otimes V,W)), \\
\psi:{_B\mathsf{Hom}}(U,{_B\mathsf{Hom}}(B\otimes V,W))\to{_B\mathsf{Hom}}(U\otimes V,W)
$$
given by
$$\varphi(f)(u):a\otimes v\mapsto f(a\cdot u\otimes v),\\
\psi(g):u\otimes v\mapsto g(u)(1\otimes v),$$
respectively, for all $u\in U,v\in V,f\in {_B\mathsf{Hom}}(U\otimes V,W)$ and $g\in {_B\mathsf{Hom}}(U,{_B\mathsf{Hom}}(B\otimes V,W))$. Then check that these are inverses each other. $\square$
Proposition: For a Hopf algebra $H$ with antipode $S$ the functor ${\mathsf{Hom}}(V,-)$ is right adjoint to the functor $-\otimes V:{_{B}\mathsf{Mod}}\to {_{B}\mathsf{Mod}}$. The left $B$-action on ${\mathsf{Hom}}(V,W)$ is given by
$$(b\cdot f)(v) = \sum b_1f(S(b_2)v).$$
Proof: Denote by a bullet $\bullet$ the given action on a module. For example, ${_\bullet V}\otimes {_\bullet W}$ denotes the diagonal left action on $V\otimes W$. Since $H$ is a Hopf algebra, the canonical map
$$\mathfrak{can}: {_\bullet H}\otimes V \to {_\bullet H}\otimes {_\bullet V}, \quad h\otimes v\mapsto \sum h_1\otimes h_2\cdot v$$
is an isomorphism of left $H$-modules with inverse
$$\mathfrak{can}^{-1}: {_\bullet H}\otimes {_\bullet V} \to {_\bullet H}\otimes V, \quad h\otimes v\mapsto \sum h_1\otimes S(h_2)\cdot v.$$
If you plug it into ${_H\mathsf{Hom}}(H\otimes V,W)$ you realise that
$${_H\mathsf{Hom}}({_\bullet H}\otimes {_\bullet V},{_\bullet W})\cong {_H\mathsf{Hom}}({_\bullet H}\otimes V,{_\bullet W})\cong {\mathsf{Hom}}(V,W)$$
(recall that for any algebra $A$, $A\otimes -:\mathsf{Vec}_{\Bbbk}\to {_A\mathsf{Mod}}$ is left adjoint of the forgetful functor). Now, if you induce via this isomorphism the left $H$-action on ${\mathsf{Hom}}(V,W)$ that comes from the left $H$-action on ${_H\mathsf{Hom}}({_\bullet H}\otimes {_\bullet V},{_\bullet W})$ of the previous lemma, you will find the one in the statement. $\square$
Observe that for the group Hopf algebra, $\Delta(g)=g\otimes g$, $S(g)=g^{-1}$ and you recover the situation of your first example. For the universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ you have that for every $x\in\mathfrak{g}$, $\Delta(x)=x\otimes 1+1\otimes x$, $S(x)=-x$ and you recover the second example.
Remark: It deserve to be noticed that the lemma holds true also for quasi-bialgebras and bialgebroids (for bialgebroids, see for example Proposition 3.3 in P. Schauenburg, Duals and doubles of quantum groupoids ($\times_R$-Hopf algebras), New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, 273–299.).
Remark: I would like to point out finally that indeed this provides a characterization of Hopf algebras: an algebra $H$ is a Hopf algebra if and only if its category of modules is closed monoidal with monoidal underlying functor $\omega:{_A\mathsf{Mod}} \to \mathsf{Vec}_{\Bbbk}$ preserving internal homs.
