I haven’t even heard about elementary OS until now. Also, the thumbnail misspells Danielle, so I looked up her name and it turns out she’s trans, and now it seems like the misspelling of her name was intentional
Edit: another user said this already, but you should probably CW this for transphobia, some of the comments under the video are really unhinged
I know you were quick to dismiss inner product spaces, but it might work if we define the inner product to be the “similarity” between the genders. The definition of an inner product space could be met, if addition and scalar multiplication of gender are defined carefully. Addition of genders would represent being both genders at the same time, and scalar representation would reflect the intensity. In this way, addition would be an average with the intensity doubled. So the more intense a gender is, there are more similarities with a similar gender, and more differences with an opposing gender. If the genders are similar, this would be positive, negative if significantly different, and zero in the case of the genders not being opposing nor similar. Since inner product spaces are normed vector spaces, making this a normed vector space would follow the triangle inequality. Although you do make a point about vector spaces being problematic because they use positive and negative numbers, which does in a way reinforce binaries. Although if it was infinite dimensional I don’t think this would be too much of a concern. I think it was yewler who suggested using a vector space of real valued functions over some set, this seems like something to consider.
I’m not sure if this helps, this is really just speculation on my part.