I took a quick look into Timmermann's book "An invitation to Quantum Groups".

It refers to:

Saad Baaj; Georges Skandalis
<i>Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres</i>
Annales scientifiques de l'École Normale Supérieure (1993)
Volume: 26, Issue: 4, page 425-488
ISSN: 0012-9593

They describe finite dimensional C${}^*$-Hopf algebras in terms of multiplicative unitaries.
Theorem 2.2 shows that if you have commutative multiplicative unitaries you get a locally compact group and therefore in the finite case, a finite group.

This gives the first statement. The second statement is equivalent to the first by duality.

\*edit*

Such a result is already stated as Theorem 3.3 in

L. I. Vaĭnerman and G. I. Kac, <i>Nonunimodular Ring Groups and Hopf-Von Neumann algebras</i>,
Mathematics of the USSR-Sbornik, Volume 23, Number 2