I'll try to answer your question, sorry if it doesn't satisfy you

Rather than a "3rd gender", eunuchs are more fitting to be described as a "man who never reaches puberty" or if you want to be more technical, "a subversion of societal expectations for masculinity."

You see, the castration of eunuchs is very much important in defining what a eunuch is. In most Islamic societies, the whole process of a child reaching puberty or masculinity (baligh) came with several expectations, namely the ability to father children, the growth of facial hair, and the changing of voice. Castration pretty much removed a person's capability to attain such things. This meant that a eunuch is pretty much an arrested male gender; a child that never reaches puberty with all the androgyny that young boys can exhibit.

This much is pretty intentional. The palace eunuchs were intended to occupy an intermediate asexual space of the harem, separate from the male harem Third Court and the female harem. The eunuchs were allowed to enter the male part of the Palace because they are male but were also allowed to enter the harem because they're not fully adult. Indeed, only those who were not considered to be fully adult males were permitted in the inner worlds of the palace: in the male harem household, boys and young men, eunuchs, dwarves, and mutes; and in the family harem household women and children. The eunuch freely moved between the world of men and the world of women just like how a young child could move between different physical and moral spaces without disturbing the moral order because they both have not reached adulthood, and in the former case, will never ever be.

The space eunuchs occupied could also be understood as the intermediate between masculinity and femininity, which lends them the ability to freely enter both the world of men and the world of women. Premodern Islamic medical discourse did not properly define a gender binary. Instead, men and women were viewed as part of the same continuum; Man in this scheme of things was the crowning achievement of terrestrial creatures, whereas woman was regarded as a less-developed version of man. The castration of a eunuch made them stopped being fully masculine or fully feminine, as children were.

And this perhaps is the way people also saw them in terms of gender: a perpetually young child, or an imperfect man.

For readings I highly recommend Jane Hathaway's The Chief Eunuch of the Ottoman Harem and Leslie Pierce's The Imperial Harem.