Tx,
animal biology is so varied that it is unsafe to generalise. Certainly most female insects that I can think of are mute, but then most male insects I can think of are apparently mute too. You also are correct in saying that their auditory signals are not vocal, but then vocalisation is such characteristically vertebrate behaviour that to insist upon it would be somewhat chauvinistic, coming from a vertebrate. Certain Tenebrionid beetles known in South Africa as "toktokkies” (and possibly endemic, for all I know) signal to each other by knocking their abdomens on the ground. I do not know whether one could tell their sexes apart by their knocks but I suspect that the beetles can. Female mosquitoes attract males by the note of their wings, which is distinct from the note of males.
You spoke about birds and mammals calling, not singing. It is not a clear distinction. I am not at all sure offhand of any species, possibly, but by no means certainly, excluding humans, where I would be willing to answer for that distinction. Just the other day two blue cranes flew over, probably a pair, their calls alternating in a grating purr: "krrrrr…krrrrr…” The voice of one was distinctly deeper than the other.
The male of our local Spotted Eagle Owl says: "Hoo hooooo”. The female says: "Hooo hoo hooooo”. Our antiphonally singing bokmakierie shrikes commonly sing in duets, sometimes very complex duets. I am just short of completely unmusical, and while I cannot speak for other people, I certainly cannot tell the sexes apart by their voices. Then again we have a lot of Egyptian geese locally and you cannot possibly mistake the sexes' voices. The male has a rather husky note, subdued except when issuing territorial challenges, when he becomes markedly raucous. The female has a persistent, petulant, loud, unmusical quack. Whether she is calling her brood, challenging intruders, or just plain kvetching, she goes on and on and on.
Cats have rather complex vocal repertoires that vary genetically, but for example when a female cat is in oestrus and "calling" she does not sound like anything or anyone else, especially if she is Siamese or Burmese. Conversely, a tomcat exchanging pleasantries with rivals sounds very different.
I hope this is helpful in answering your question, but I also hope that I am beginning to convey some of the reasons why I am reluctant to generalise.
Cheers,
Jon