Why do people conflate being gay with the "trans" movement?
Because in the minds of many they are all freaks, perverts and weirdos. It would be a rare individual that supports the gay community's efforts to be accepted but rejects the transexual community efforts to do the same.
When gay people fought for equal rights, such as the right to marry, they were not infringing on anyone else's rights. In other words, in giving them equal rights there was no conflict of interest with another group.
This is not so with gender identity. Firstly it is something no one is able to define. It is classed as a 'feeling of discomfort' around ones gender. But how can a biological male know what it feels like to be a woman?(spoiler alert: no one does. Even women don't 'feel like' women.If we didn't have bodies and were just floating heads no one would know what gender they were).
Perhaps it would be easier if you added the word "roles" after gender. It may help you understand. As I understand it, those who feel they are pretending when they are pressured by society to project a feminine or masculine appearance and characterizations want to stop pretending. They are more comfortable projecting the opposite of what they were told they should. We used to call them tom boys or femboys, butch,poofers and pansys. They are asking us not to but just address them the way they want.
Secondly it does infringe on the rights of others. On the right of women to feel comfort, safety and dignity in their intimate spaces.
Both does that cut both ways.