Why is $0$ excluded in the definition of the projective space for a vector space? 
For a vector space $V$, $P(V)$ is defined to be $(V \setminus \{0 \}) / \sim$, where two non-zero vectors $v_1, v_2$ in $V$ are equivalent if they differ
  by a non-zero scalar $λ$, i.e., $v_1 = \lambda v_2$.

I wonder why vector $0$ is excluded when considering the equivalent classes, since $\{0\}$ can be an equivalent class too? Thanks!

You could do this, but the resulting space would not be as useful.
For example, suppose $V$ is $\mathbb{R}^n$ equipped with its usual topology.  Then the projective space $P \mathbb{R}^n$ can be made into a topological space by giving it the quotient topology.  If you include 0 as in your suggestion, the projective space would not be Hausdorff in this topology; in fact, the only open neighborhood of the equivalence class $\{0\}$ is the entire quotient space.
