Let X be a Banach algebra and $x,y\in X$ by definition then
$\Vert xy\Vert\leq\Vert x\Vert \Vert y\Vert$
and the intuition could be that it makes multiplication continuous, which is a nice property.
Is there an similar intuition behind the $C^*$-identity? I know it implies that the involution is an isometry, but if that was what was wanted it would seem more direct to just assume that.
Adjoint is an operation which depends on the inner-product. Adjoint is not a purely algebraic concept; nor is it purely topological. Adjoint depends on the inner-product. A non-unitary change of basis for square matrices changes the adjoint, and you can work out what that looks like. You could study an operator algebra on a Hilbert space where you use an adjoint involution generated by one inner-product, and an operator norm generated by another topologically equivalent inner-product.
The $C^{\star}$ identity becomes a consistency condition between the inner-product used to generator the operator norm and the inner-product used to generate the involution.