This is consistently true, for separable $G$.

In a recent preprint, Hrusak and Ramos-Garcia (http://www.matmor.unam.mx/~michael/preprints_files/Frechet-malykhin.pdf) have produced a model of ZFC, by iterating a Laver-type forcing, where every separable Frechet group is metrizable, and in particular, the product of Frechet groups is metrizable, and thus Frechet.

I haven't read the paper, but Justin Moore pointed me to it when I asked him about this question.

This is consistently true, for separable Frechet $G$.

In a recent preprint, Hrusak and Ramos-Garcia (http://www.matmor.unam.mx/~michael/preprints_files/Frechet-malykhin.pdf) have produced a model of ZFC, by iterating a Laver-type forcing, where every separable Frechet group is metrizable, and in particular, the product of Frechet groups is metrizable, and thus Frechet.

I haven't read the paper, but Justin Moore pointed me to it when I asked him about this question.