cis and trans would still exist
no, because Gender (not physical sex), is a Social Construct. Trans is only an issue because of societies expectations of Gender.
People don't feel free to express what they want to be like. Various Gender categories are not even legally recognised in most countries.
If people could behave, dress, have what ever body modification surgery they wanted (including transplants), with no expectations from society based on physical chromosomes, no segregated toilets, toys, colours, clothes etc, then you could imagine a society were someone would have difficulty explaining what Trans is, or any of the other "non-cis or non-straight" orientations, or indeed why cis-straight" orientation has any significance. Imagine a society were producing egg or sperm or neither, or deciding to swap was of no more interest to people than if people chose to pierce their ears, or had brown rather than black hair*.
No-one chooses their own chromosomes. How we feel about our own identity and how we would like to live and express it doesn't seem to be something that can be chosen either. We can only chose how we behave, how we treat others etc, not who we are. Societies decide on what are normal "identity" aspirations or beliefs and if you don't fit that it's a constant stress.
Physically we are dimorphic due to chromosomes (and hormones based on genes), so there are 1/5000 females with no womb, women that have different hormone levels and grow male like hair, males with more female characteristics due to hormones, (everyone in sense starts off female and differentiation is later, males can lactate). Some people are hard to physically differentiate at birth, visually. But your identity or the way you are attracted to others can ignore all of this. We don't know why, and it's only really a problem for people because of societies' gender stereotypes and expectations based on physical birth sex (which actually isn't even perfect or clear for everyone). In additional complication, a "trans" person might be attracted to either sex, no sex or both sexes, it's not about what other people they are attracted to, it's a separate issue to "straight", "gay/lesbian" or bisexual. It's not the same either as people that want to "cross dress", they may feel they somehow got issued the wrong body.
I'm not going to write a story specifically with these issues, because it's too complicated, too many misunderstandings and even LGBT is misleading daft umbrella simply lumping everyone with disparate outlooks etc together as they are all supposedly minorities. Perhaps also there is spectrum of all these aspects, rather than simple portrayal in media.
EDIT
[* Changing hair colour is generally quite acceptable to most people. But some condemn it. People may even darken or lighten their skin or curl or straighten hair]