Could someone help me find the fallacy of my argument :
Let V be the universe of all sets, let M be a countable transitive model of ZFC and let $M[G]$ be a generic extension. Let $\alpha, \beta \in M$ be ordinals. Since $M\models ZFC$ and $M$ is transitive, $$\alpha, \beta \in M \; \Rightarrow \; \alpha \times \beta \in M \; \Rightarrow \; Pot(\alpha \times \beta)\in M \; \Rightarrow \; Pot(\alpha \times \beta) \subseteq M,$$ (where $Pot(\alpha \times \beta)$ is the powerset of $\alpha \times \beta$), in particular every function $\alpha \to \beta$ that exists in $V$ is an element of $M$. So every function $\alpha \to \beta$ in $M[G]$ must also exist in $M$. In particular, $|\beta|^M = |\beta|^{M[G]}$, what contradicts the idea of collapsing cardinals.
You're missing the key point here. The power set operation is relative to $M$.
Indeed, $M$ is a countable model, so it does not contain all the power set of $\omega$, let alone any other infinite ordinal, since those are uncountable. Instead it only contains a fragment of this power set. But from the point of view of $M$, the important thing is that there is a set which collects all of the subsets of $\omega$, or $\alpha\times\beta$, known to $M$.
Since $M$ is countable, this "so-called power set" is also countable, which leaves us room to find new functions which are generic over $M$.