Both the statement are true, which is fine because we talk about relative consistency here.

If $\sf ZFC$ is consistent then we know how to generate a model of $\sf ZFC^-+\it G\neq V$.

On the other hand, it is consistent that $G$ is generated by power sets iterations from a countable set of "atoms" of the form $x=\{x\}$. Then we can easily generate a permutation model in which this set of atoms is not well-ordered, but every well-founded set is well-ordered.

Indeed we can even require that $|V|=|\sf Ord|$, but then $|G|\neq|V|$ because $G$ cannot be well-ordered.

Both the statement are true, which is fine because we talk about relative consistency here.

If $\sf ZFC$ is consistent then we know how to generate a model of $\sf ZFC^-+\it G\neq V$.

On the other hand, it is consistent that we have a proper class of atoms: sets of the form $x=\{x\}$, with global choice. Then by taking a class sized permutation model we can ensure that the class of the atoms is not well-orderable, while the axiom of choice for sets holds, and global choice for well-founded sets holds.

In that case we have $G\models\sf ZFC^-$ but $|G|\neq|V|$.