Most attention to the Swahili city-states emphasizes how cosmopolitan they were, as a meeting point for the Indian Ocean-Arabian-African interior trade. What kind of native environment did that breed? What was the experience of childhood, attitudes towards children, children's activities (education, work, play) in a multicultural medieval world? How were things different for boys and girls? Above all, what was distinctive about growing up in (Islamic) Zanzibar versus Baghdad or Damascus?
That is an excellent question!
Unhappily, the stuff that I am most conversant in (archaeolgical texts and the Kilwa chronicle) have very little to say on the subject. I can only give you some preliminary notes to guide you so far.
Digging into SOAS's swahili manuscripts project page does turn up an untranslated copy of the Utenzi wa Mwana Kupona, a poem of motherly advice to a daughter. I have turned up a literary essay that contains some translations, which I will link below^1. However, this poem is not really in keeping with with your requested Medieval time-period, since the poem was composed in the last decades of the 19th century.
The World of the Swahili by the antropologist John Middleton can give you a good deal of information about marriage, family, inheritance, etc. However, this is all information from 20th century Swahili merchant society. Obviously, we can't assume that his observations can be securely carried backwards into the medieval era.
Frankly, I should do quite a bit more searching before I can give you much more information. Watch this space.
1 www.iiste.org/Journals/index.php/JEP/article/download/9919/10124
Warning- that is a direct link to a PDF download of the article.