The Sapir-Whorf hypothesis states, briefly put, that linguistic structures affect cognitive processes. I am interested in finding out how much is known about the development of gender identity from this point of view. That is: Do the gender-related structures of each language influence the way children develop such an identity? Ideally, I would like to know about recent empirical studies comparing the development of gender identity in, say, Semitic languages (heavily gendered), Indo-european languages (gendered) and finno-ugric languages (completely genderless, as far as I know). I am aware of some works in these field, and also about some other very interesting empirical works in relation to Sapir-Whorf hypothesis, like Everett's field work on Pirahã language or this work about absolute spatial description in the Yimithirr's language, but I have not found anything similar (empirical) concerning recent studies on the construction of gender identity for speakers of families of languages which treat (very) differently the gender.
One fascinating example (perhaps completely irrelevant in a linguistic discussion, though, but at least thought-provoking) that I have in mind is the existence of third gender or inter-gender persons, such as the Mahu of Hawai'i and Tahiti or the fa'afafine of Samoa, which were common in pre-colonial polynesian cultures, which precisely possess genderless languages. Could this somehow, to some degree, be related to the language and be related to linguistic relativism? Has this been researched?