What is specifically different for transgender people is that the relevant difference in question is often their genitals—and not even necessarily the genitals they currently have, but the genitals they were born with. Regardless, genitals. And no one wants to hear about a stranger's genitals. That's the difference.
The top of my head is balding, I have a full beard, my chest is flat, I speak with a voice in the men's register, and I dress similarly to other men in an unremarkable way. If I go to a movie theater, and I ask the person working at the concession stand, "Where's the bathroom?," that person will say, "The men's room is over there." My response is: "Great, thanks." Never, ever, ever would I respond: "Did you know I was born with a vagina?" Because the person working at the concession stand does not care. I would arguably be sexually harassing other people if I brought it up. (I sort of feel that I am being gross to you now, even though you directly asked me a question to which I am giving the correct answer. This is one way in which transgender people are marginalized: people ask us questions that they are perfectly able to think through for themselves but they seem to be gratified when we debase ourselves by explaining our genitals to them.) Nor would I ask if the movie theater has a third/fourth bathroom for transgender men and women, because obviously it does not. No public building I've ever been in, or have ever known of, has bathrooms marked "Cisgender" and "Transgender." (Sometimes there is a separate, non-gendered bathroom designed for people who use wheelchairs. That might serve the needs of someone who wants a non-gendered bathroom for reasons of gender, though ideally they would not take up too much time/space in the wheelchair-accessible bathroom because it needs to remain free for people who need it because of a physical disability.) I would not petition the movie theater chain to spend hundreds of millions of dollars upgrading all their cinemas to have transgender-specific bathrooms because I do not even want or need to be in a transgender-specific bathroom. (I do, after all, know how to use the men's room, and that feels like the right place for me to be, and no one in 23 years has ever tried to kick me out of a men's room, so I guess the feeling is mutual.) And, while some transgender people prefer gender-neutral bathrooms, I've never heard anyone say they want to be in a transgender-specific bathroom. If there were such a thing as a transgender-specific bathroom and if I entered it, it would indirectly announce my birth genitals to everyone in the theater, which is Too Much Information that they probably didn't want to know. Also, it would be unsafe for everyone who went in there. If there were transgender-specific bathrooms, haters would lurk outside/inside them to identify all the trans people, "out" them on the internet or to their employers, or even violently attack them, right? If you were eating alone in a restaurant after 11 p.m., and a group of men at the bar were watching you, would you feel safe if they saw you open a bathroom door marked 'Transgender'? There is no real benefit for any trans person to open a door marked "Transgender People Only," especially if they don't want to be "out" as transgender to everyone around them at that moment. Nor is there any benefit for other people, who do not care (or shouldn't care) what genitals their neighbors and acquaintances were born with.
If you see a problem for movie theater patrons who are spending merely three hours in the building, consider how it would play out at a workplace. You think your boss needs to know your genitals at birth and whether you have different genitals now? When should you share that with your boss? During the job interview? Is this question going to be asked of everyone? Or is it only transgender people who are responsible for bringing it up, perhaps by listing it on their résumé?
Or when you are introducing people at a party? If you've just met Tim, and you say, "Hey, Jennifer, I'd like you meet this cool guy. His name is Tim," it is not Tim's responsibility to say, while he's shaking Jennifer's hand, "Nice to meet you, Jennifer. I have a vagina, so I'm not exactly a 'guy.' You should probably call me 'hormonally enhanced person' to be more factually precise, and you should use the pronoun 'trans' instead of 'he/him/his,' please, as in: 'Trans name is Tim.'" Jennifer had been planning to say, "Well, my ride's here, gotta go," so she did not need that information. And, even if she ends up knowing Tim for years, she may never need that information.
On the other hand, maybe Jennifer is diabetic, so she is also hormonally enhanced, and maybe she had surgery on her vagina following childbirth, and in these respects maybe she and Tim have something in common. But maybe they could find that out in a way that evolves more naturally, if they end up being friends, since these are details about their medical histories. It isn't Jennifer's responsibility either to greet people at a party: "Hi, I'm Jennifer, I'm diabetic and I've given birth vaginally." No one asked for that information.
If the people at the party want to be grouped by what baseball team they cheer for, "cisgender" and "transgender" are factual but irrelevant categories.
When you say (at the end of your comment) that you don't want to enter "white men's spaces," I don't know specifically what real-life situation you're referring to. If you're talking about doors marked "White People Only," that is a racial segregation scenario on which I am not well positioned to comment, in part because I am white and haven't lived though racial segregation and in part because I don't know where your analogy is going. What I do know is that, today, in real life, in the societies I'm familiar with, buildings typically have only two bathrooms marked "Men" and "Women," and gyms definitely have only two locker rooms, and therefore transgender people find ourselves in the position of needing to pick one or else not use the facility and just go home. Some people suggest better options: gender-neutral facilities in certain scenarios, for example. Maybe those are good ideas, but I can't review them all now, and bathroom remodeling isn't my area of expertise. But I can tell you what's a bad, unhelpful idea: That transgender people ought to publicly announce their genitals, their personal lifelong history of how they exist in their gender, and/or their private feelings about their gender and their body while they are in the middle of a movie theater lobby, when no one in the movie theater is asking nor cares. If a transgender person is happy using the men's or women's room, then that's where they belong, and, especially when no one is the wiser for it, there is no complaint to be raised. Nor does flagging and outing trans people as trans (or requiring them to out themselves as trans) serve some lofty philosophical purpose of reflecting "truth," since it doesn't get cis people any closer to understanding what trans people "really" are. If cis people really want to understand trans people, they can start by listening to trans people—deeply, carefully, for a long time, forever, really—rather than deciding in advance what the correct answer is and then browbeating trans people into echoing the predetermined "correct" answer.
