Here the definition of Banach star algebra is given as Banach algebra with an involution. In the book by Murphy for example, it is given as Banach algebra with an involution plus the condition that $\|a\|=\|a^\ast\|$.
My question is: is the condition necessary or does it follow from the other conditions?
The property does not follow from the others, but is useful and satisfied by enough examples to make it often worth assuming. Aside from C*-algebras, $L^1$ algebras on locally compact groups satisfy this definition. One of the useful consequences of the property is that it implies that $*$-representations of Banach $*$-algebras on Hilbert space are (weakly) contractive.
You can consider various norms on $M_2(\mathbb C)$ with the usual operations to see that the other axioms can hold without this one.