As an example, take the Virasoro algebra, i.e. $V$ is spanned by elements of the form $L_{-2}^{k_1} \cdots L_{-n}^{k_{n-1}} \Omega$ where $\Omega$ is the vacuum and $n \geq 2$. As I understand, we define
$$C_2(V) = \{\psi^i_{-h_i-1}v | v \in V\}$$
where $i$ labels all the operators strongly generating the VOA, and $h_i$ are the conformal dimensions. So, for the Virasoro algebra one has,
$$C_2(V) = \{L_{-3}v | v \in V\}.$$
Now it is claimed that $V/C_2(V) \cong \mathbb C[x]$, where the isomorphism maps $L_{-2}\Omega \mapsto x$. However, I do not see how this is the case. We are modding out all elements of the form $L_{-3}v$, i.e. all those that contain a factor of $L_{-3}$. Then say, $L_{-10}\Omega$ cannot be part of an equivalence class with a representative given by a power of $L_{-2}\Omega$, so I do not see how $V/C_2(V)$ is just generated by $L_{-2}\Omega$.
For reference, see page 14 of Arakawa's paper.