I would associate N/A more with agender (someone who has no gender) more than someone who has a gender but it is not listed. For me, my gender is applicable, it's just not in that list.
"Not Listed" might be a better title for N/A?
Edit: Just saw your edit, makes more sense. Still feels a little on the edge for me, but we're all going to have different needs out of something like this.
A very very very extremely minor adjustment might be to rename "other" to "not listed".
The reasoning is that the word "other" can be alienating to some folks. The way we use the phrase "the other" to imply something foreign and untrustworthy in common language is a good example of why someone might not want to be called "other" in a form.
Now, this is a really minor thing, and won't impact 99.9+% of your users. But it could impact a few people! It's ultimately up to you whether you think it's clearer/more appropriate for your app to use "other", "not listed", or something else! My goal is not to be prescriptive, just to provide a little color and reasoning.
Where are you from that your nation doesn't have some concept of gender (differing societal roles for, e.g., men and women)? I'm genuinely asking because I think it would be interesting to learn more about the language and culture.
1) Because that's how people do it. Sometimes we all collectively settle on a standard that isn't the best, but that's just how it's done.
2) It allows for more easily expanding to multiple pronouns, e.g. "he/they" or "she/they", which indicate that the person is ok with "she/her/hers" or "they/them/theirs".
3) It makes it easier to state neopronouns (e.g. xe), by standardizing the form -- he/him/his is the same format as sie/hir/hirs or xe/xem/xems.
4) (My opinion) It feels more natural to say "he him" than just "he". "What are your pronouns?" "He." I dunno, it's a short enough utterance that it could be easily to mishear?
What you've described is what most non-binary, trans, and many queer folks believe. (And many allies as well.)
The idea that there are "only two" genders is, imo, a bit of a 20th century lens, and I expect we'll move to disentangle gender from what anatomy do you have.
I agree that asking for gender is fraught. I am non-binary, and I constantly have to either lie or just choose "Not sharing". The best systems just take a string and leave it at that.
Gathering this by employers is part of US federal law. It is also passed to your insurer, and used to do... insurance things invisibly.
Gathering this by companies is so they can better profile you and serve you tailored content (as defined by them.) E.g. If you respond "Man", they might want to send you fewer ads for tampons and more ads for razors or whatever. Note, I'm not saying this is a good thing, but it's the reasoning.
That's the opposite of what the parent was suggesting I think?