Let us fix $M$ a model of $ZF$. Inside $M$ we can build the von Neumann hierarchy starting with $V_0, V_1, etc.$ and then $V_{\alpha}$ for any ordinal $\alpha$.
What is the "status" of that ordinal $\alpha$ ? Is it the $\alpha$ of the model $M$, that we could note $\alpha_M$ ? Or is it the $\alpha$ of the "ambiant set theory", or let's say a naïve meta-mathematical ordinal ?
One builds the Von Neumann hierarchy in $M$ using ordinals in $M$, so we must have $\alpha \in M$ and that $M \models \alpha$ is an ordinal. $\alpha$, as ane element of the universe $V$ (or "ambient set theory") need not be an ordinal, but that's irrelevant as long as it is an ordinal in $M$.