As Yemon says, it is the Arens product on the double dual of a Banach algebra that you're thinking of. Yes, $A^{**}$ and $\pi(A)''$ are isometrically isomorphic as Banach spaces and $*$-isomorphic as C*-algebras. This is surely in Sakai's book that you're already looking at.

A related question that you didn't quite ask is whether two C-algebras which are isometrically isomorphic as Banach spaces must be $*$-isomorphic as C-algebras. The answer is no --- there are C-algebras which are not $*$-isomorphic to their opposite algebras (change the order of the product and leave everything else, including norm, alone). However, isometric C-algebras will have the same state space and this implies that they are Jordan isomorphic, and roughly speaking, switching the order of the product is the worst thing that can happen. See the book State Spaces of Operator Algebras by Alfsen and Schultz.

The "right" version of this result is that two C*-algebras that are completely isometric must be $*$-isomorphic.