I think the author accidentally described the dual of the Hopf algebra you're thinking of. Finite group rings are usually endowed with multiplication $(g,h)\mapsto gh$ and comultiplication $g \mapsto g\otimes g$ (see here).

The coordinate ring $k[G]$ is obtained by dualizing. Then $g \mapsto g\otimes g$ becomes $e_g^2 = e_g$, where $e_g$ is the function on $G$ that maps $g$ to $1$ and all other group elements to $0$.

I think the author accidentally described the dual of the Hopf algebra you're thinking of. Finite group rings are usually endowed with multiplication $(g,h)\mapsto gh$ and comultiplication $g \mapsto g\otimes g$ (see here).

The coordinate ring $k[G]$ is obtained by dualizing. Then $g \mapsto g\otimes g$ becomes $e_g^2 = e_g$, where $e_g$ is the function on $G$ that maps $g$ to $1$ and all other group elements to $0$. Comultiplication will look exactly the way you described it (i.e. $e_g \mapsto \sum_h e_{gh^{-1}}\otimes e_h$).