For $X$ and $Y$ normed vector spaces with product space $(X\times Y,\|\cdot\|_X+\|\cdot\|_Y)$ and continuous dual spaces $X^\prime$ and $Y^\prime$, I know that $((X\times Y)^\prime,\|\cdot\|_{(X\times Y)^\prime})$ is isometric isomorphic to $(X^\prime\times Y^\prime,\max\{\|\cdot\|_{X^\prime},\|\cdot\|_{Y^\prime}\})$ via the isometry $$J\colon X^\prime\times Y^\prime\to(X\times Y)^\prime,~(J(f,g))(x,y):= f(x)+g(y).$$ Is $J$ also an isometry between $((X\times Y)^\prime,\|\cdot\|_{(X\times Y)^\prime})$ and $(X^\prime\times Y^\prime,\|\cdot\|_{X^\prime}+\|\cdot\|_{Y^\prime})$? I tend to say no, but can't come up with a suitable counterexample because of the structure of the dual norm.
Can there exist some other isometric isomorphism between those two spaces?