I think Jessica Valenti's point is that women candidates come in for additional gender specific questioning of a type that men candidates don't. They will be questioned about "women's issues" in the way that women journalists and columnists (in the MSM) are expected to focus on women's issues, freeing up time for men journalists to write about "weightier" topics like economics.
E.g., while the primaries were still ongoing and after the Reade allegations against Biden re-effervescenced, only the remaining women were asked about their thoughts on the matter, but the men candidates were not, as if sexual harassment were purely a women's issue. Little scrutiny was brought to bear on Gore, Lieberman, Biden or HRC's whatisname as VP candidates independent of the corresponding presidential candidates' policies and they were rarely asked about previous conflictive stances. Because the VP candidate will be a woman, she will be asked about her previous advocacy or policies in conflict to Biden's on women's issues, in a way that neither men VP candidates for men POTUS candidates nor whatisname 2016 Democratic VP candidate under HRC have been questioned on "men's issues".
The misogyny that JV refers to is also visible in the way that Republican women in the Twump administration are taken to task over "separation of immigrant children" as if it were particularly horrific that a woman condoned it but only something to be expected from a man.