If we look to models of $\small \sf ZFC$ we can have two transitive models $N,M$ such that for some formula $\phi$ in the language of $\small \sf ZFC$ we have:
$\forall \mathcal M ([\mathcal M\models ZFC] \to \mathcal M\models \exists!x (\phi))$
$M\models \exists x (\phi \land |x| = \kappa)$
$N\models \exists x (\phi \land |x| \neq \kappa)$,
for example left $\phi$ be $x=P(\omega)$ and let $\kappa= \omega_1$. I'll refer to the above as "size statements about some sets theorized to uniquely exist by $\small \sf ZFC$, are not absolute".
Now examine this theory, the question is:
Can we have the above situation raised with it also? In other words, can it theorize the unique existence of some sets the size statements about which are not absolute?