Trevor pointed out, that
there are models of $\mathsf{ZFC}$ in which every set is definable. In this case the collection of definable sets of the model is equal to the universal class of the model, which of course is not a set of the model.
That means: the collection of definable sets of a model can be a proper class, i.e. doesn't have to be a set.
But can the collection of definable sets – definable or not – be (co-extensive with) a set?
Added (following Carl Mummerts advice):
A set is a member of a set-theoretic universe, i.e. an element of a model $\mathcal{M}$ of a set theory $\mathsf{ST}$, e.g. $\mathsf{ZFC}$.
A set $x$ is definable if there is a finite formula $\varphi(y)$ in the first-order language of (any) set theory – with signature $\sigma = \lbrace \in \rbrace$ – such that $x = \lbrace y : \varphi(y)\rbrace$, i.e. $(\forall y)\ y \in x \leftrightarrow \varphi(y)$.
What I am - admittedly - unspecific about is whether $(\forall y)\ y \in x \leftrightarrow \varphi(y)$ means
there is a model $\mathcal{M}$ of a set theory $\mathsf{ST}$ with $\mathcal{M} \models (\forall y)\ y \in x \leftrightarrow \varphi(y)$
for every model $\mathcal{M}$ of a set theory $\mathsf{ST}$ it holds $\mathcal{M} \models (\forall y)\ y \in x \leftrightarrow \varphi(y)$
$\mathsf{ST} \vdash (\forall y)\ y \in x \leftrightarrow \varphi(y)$
So for a given set theory $\mathsf{ST}$ my question is threefold.
The question arises what a set theory $\mathsf{ST}$ is supposed to be. I assume: its signature $\sigma$ is $\lbrace \in \rbrace$ and its axioms include the axioms of a naive set theory.