Yes it is true. But not for the commutative hopf algebras in general. You need some more assumptions: first we need the field to be algebraically closed and of characteristic zero. You also need cocommutativity and finite dimensionality of the hopf algebra to have a full "correspondence".To be more precise:

it can be shown that there is an equivalence of Categories, between the Category of commutative, cocommutative, finite dimensional Hopf algebras $\mathcal{H}$ (over an algebraically closed field, of characteristic zero) and the category ${\mathcal{A}\mathcal{b}}_{fin}$ of the finite, abelian groups. It is possible to construct fully faithful and essentially faithful functors between $\mathcal{H}$ and ${\mathcal{A}\mathcal{b}}_{fin}$.

Start from an object of $\mathcal{H}$ i.e. a commutative, cocommutative, finite dimensional Hopf algebra $\mathcal{H}$, over an algebraically closed field, of characteristic zero. The set $G(H)$ of its grouplike elements forms a finite abelian group i.e. an element of ${\mathcal{A}\mathcal{b}}_{fin}$. It is relatively easy to see that a hopf algebra morphism induces an abelian group homomorphism.So you get a functor

$\mathcal{G}:\mathcal{H}\Rrightarrow {\mathcal{A}\mathcal{b}}_{fin}$

On the other hand, start from a finite abelian group $G$ and take its group hopf algebra $kG$. It is clearly commutative, cocommutative and finite dimensional, i.e. an object of $\mathcal{H}$. On the other hand, an abelian group homomorphism induces -by linear extension, due to the universal property of the group algebra- a morphism of hopf algebras between the corresponding group hopf algebras.So you get a functor

$\mathcal{F}:{\mathcal{A}\mathcal{b}}_{fin}\Rrightarrow \mathcal{H}$

Now, it can be shown that:

$\begin{array}{cccc}\mathcal{G}\mathcal{F}=I{d}_{\mathcal{A}{b}_{fin}}& & & \mathcal{F}\mathcal{G}\cong I{d}_{\mathcal{H}}\end{array}$

Consequently, the functors $\mathcal{G}$, $\mathcal{F}$ constitute an equivalence of the categories $\mathcal{H}$, ${\mathcal{A}\mathcal{b}}_{fin}$.

In my understanding, it is actually this equivalence of categories, which inspired the introduction of the term quantum groups (implying that the hopf algebra theory may be considered as a kind of "quantum" generalization of the group theory).

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