What I have proved is: such point exists in a (more general) convex closed set if and only if $E$ is a reflexive Banach space (One direction is proved by weakly compact. Another direction is by James theorem).
In particular, such point is unique if $E$ is uniformly convex (we may call it a projection of $x$ on $M$). Could someone present a counterexample in a more general Banach space? I will appreciate it so much.