$Φ$ is a primitive formula in the language of set theory, while $Φ^M$ is the relativisation of $Φ$ to the class $M$. I can't understand why $Φ^M = Φ$.
Let $Φ$ be $0 \in x$, it seems to me, $Φ^{\mathbb{R}} \neq Φ^{\mathbb{Z}}$, since $\{x \in \mathbb{R}: Φ(x)\} \neq \{x \in \mathbb{Z} : Φ(x)\}$. Where has gone wrong?
When we say that the relativisation of a formula $\Phi$ to a class $\mathbf M$ is something, we mean the actual syntactic object, and not some associated classt. Remember that the point of relativizations is to control where the quantified objects come from, and so it makes sense that for formulas without quantifiers, the relativization is the same as the original formula.
Given any class $\mathbf M$ the formula "$( x \in y )^{\mathbf M}$" is defined to be the formula "$x \in y$." And as we move to more complicated formulas the have recursive definitions telling us how to construct the relativisation.