Why isn't the quotient space $V/V = \{ V \}$?

If $W \subset V$, then one defines the quotient space, $$V/W = \{ v + W : v \in V \}$$

So why isn't this right?

$$V/V = \{v + V : v \in V \} = \{V \}$$?

I read that $V/V = \{ 0 \}$? Why can't the whole set $V/V$ be partition, by $V$ itself?

You are right, $V/V=\{V\}$. The reason this is sometimes written $\{0\}$ is that $V$ functions as the zero vector in $V/V$, i.e., in $V/V$, $V=0$.

$V/W$ is a vector space, and it have the zero element $0_{V/W}$.

I claim that $0_{V/W}=W$ - Indeed if $v+V\in V/W$ then $$ (v+V)+W=W+(v+V)=v+V $$

so $$ \{V\}=\{0_{V/V}\} $$

that is - it is the zero subspace.