Note: This answer discusses including biological sex on an official ID card (like a passport). This answer does not discuss including gender. I feel that a good discussion about including gender would require establishing what gender is in a universally acceptable way, which would be a large topic in itself. (Also perhaps a separate topic, as some folks might argue that biological sex and gender should both be listed in separate fields.)
tl;dr– The primary argument for including biological sex is that it's immutably associated with a person and relatively easy to recognize/verify. The primary argument against including biological sex is that some people feel offended by it, e.g. due to feeling that it violates their privacy or ability to self-determine.
Background: Biological sex is a subset of genetic profile.
Biological sex is essentially about if a person consistently expresses the genetic content associated with Y-chromosomes in their cells.(Note 1) To simplify, it's basically saying if someone has a Y-chromosome.
Unlike most of a person's DNA profile, biological sex is highly observable.(Note 2)
Main point: It's about whether noting biological sex is more useful than objectionable.
The main argument for including biological sex is that it's useful in verifying identity.
This has two common applications:
Defense against fraud.
This is part of a defense-in-depth strategy in that, while checking someone's biological sex is by itself insufficient to establish the legitimacy of their claim to an identity (e.g., as the rightful holder of a passport), it's still a major filter for little effort.
Identifying someone in a search.
If you're looking for someone, it's helpful to know their observables: stuff like height, weight, sex, race, age, hair color/style, what they're wearing, etc., can be useful. Observables are more helpful when they're more immutable and more easily observed.
I suspect that most reasonable people can agree that biological sex is useful for the above reasons; so, those who want to exclude it probably have specific objections to it beyond concerns about its utility.
I've anecdotally noticed two different types of objections to biological sex being included:
Biological sex is private information.
While biological sex is highly observable by default, some folks wish to modify their apparent sex. Having their actual biological sex noted on an official document undermines their intent.(Note 3)
Biological sex is offensive.
Some folks feel that biological sex is an inherently offensive concept. Common reasons seem to include a belief in personal control over gender identity and disdain for biological determinism (discussed in a "Tangential" section below).
Most folks don't consider biological sex to be private information. In fact, I'd guess that, among the observables typically given on an ID card, it's more common for people to feel uneasy about the fields for age, weight, or a potentially unflattering photo.
Likewise, I'd guess that most folks don't find biological sex to be offensive. However, biological sex can be seen as undermining the interests of trans-gender/sexual individuals,(Note 4) potentially even exposing them to increased risk of hate crimes.
Conclusion: It's about if utility is greater than objections.
I think that most reasonable people would agree that having biological sex on official identification cards, like passports, can have its uses. So, those who argue against its inclusion likely do so because they consider biological sex to be private information or offensive.
Tangential: Regarding disdain for biological determinism.
Some folks are free-will compatibilists. Even if you show them a machine that can predict their future decisions,(Note 5) they'd still feel like they had decision-making power.
By contrast, some folks are free-will incompatibilists. If you showed them a machine that can predict their future decisions, then they'd feel like they lack the power to make such decisions freely. They'd probably hate the machine for one of two reasons:
They'd feel that the machine is stealing their free will from them.
They'd feel that the machine reveals that they never had free will in the first place.
Biological determinism is similar. Compatibilists may feel comfortable with the concept of genetics determining their nature, while incompatibilists may feel it to be utterly dehumanizing.
Personally, I'm a compatibilist, so I think biological sex is a cool concept; that it's so powerful in its predictive ability makes it awesome. But for incompatibilists, biological sex may feel violating; like it robs them of their power to be a free person.
I note this because some folks don't like the idea that they have a biological sex that says anything meaningful about them. Assertions to the contrary can come across as an existential attack against their personhood.
Tangential: Binary sex is probably too limiting though.
Almost every scientific theory about anything is wrong. It's probably a safe bet that even our best theories of physics are somehow wrong; just, at any point in history, we're not quite sure how/why yet.
Biological sex is no exception: we know it to be an imperfect concept. This doesn't mean we should ditch it because
All models are wrong but some are useful
—Famous quote from George E. P. Box
, and biological sex is definitely a useful model.
That said, we shouldn't use a model when it doesn't work. And there're cases in which the male/female dichotomy doesn't fit, e.g. for intersex or
My point here isn't about political correctness nor about gender identity. Instead, what I'm saying is that when a person isn't objectively well-described as either male or female, forcing an ID card to say one or the other is simply printing misinformation. And that's an objectively bad thing to do.
Most biological females have XX chromosomes while most biological males have XY chromosomes. However, there're rare conditions which deviate from this norm; for example, ~0.075% of males are XXY, and ~0.005% of males are XX. The exact wording used in the above answer was selected to avoid misclassifications that can arise from assuming the basic XX/XY dichotomy.
More precisely, it's easier to connect external observables to statements about a person's DNA. By which I mean, a lot of consequences of a person's DNA can be highly observable, but the implications of many observables aren't trivially mappable to a DNA profile, making it more difficult to appreciate such associations.
One common argument for modifying apparent sex is to advertise gender. This answer is intentionally avoiding discussing gender – that'd require covering a lot more ground first, and this answer's long enough already.
This answer discusses "biological sex" as a function of genetics, not external appearance, such that a transsexual's ID card would include their physically apparent sex at birth (rather than post-transition). This isn't meant to imply anything about the nature of transsexuality nor transexuals; it's just the specific definition of "biological sex" being discussed; discussing alternatives would require extending this already-very-long answer.
Such a decision-predicting machine is often described in at least three contexts:
as a hypothetical in such philosophical thought experiments;
with respect to machine-learning algorithms predicting folks' behaviors based on, e.g., their social media interactions;
whenever people feel like they're being predictable.
I note this to help ground the concept in everyday experience, to show that it's not just some esoteric philosophical concept. For example, have you ever met someone who feels like they're not "truly alive" if someone else can guess what they'd do? Like being predictable demeans their humanity, making them less of a person? That's incompatibilism!