An identity is a multi-faceted thing which comprises not just gender identity and expression but also interests, affinities and predispositions in a melting pot which makes up who you are and it is when elements in this constitution rub against the grain of societal norms that problems arise.When trans people were either underground or transitioning in stealth, other targets invited the ire of the public. That their strangeness was more readily tolerated was due to its less offensive nature and it could be chalked up to a quirkiness which could perhaps even be ridiculed into disappearing.
Infringing on sexual orientation was the first battle and who could forget the advocacy of Anita Bryant in the 1960's which greatly mirrors the moral panics against trans people today. However playing in any way with gender is an even worse offense specifically because it violates the concept of identity in a way sexual orientation does not. Once you allow people to define themselves according to how they feel about their gender you have broken a fundamental law of western religion and its orthodoxy.
Today trans people are the most convenient target because their small population allows them to be used as fodder for what ails western culture. Conservatives can lump us in with a whole host of what they deem to be immoral ideas stemming from the evils of 20th century postmodernism. Hence, the more you allow people latitude in self identifying, the more you are attacking the underpinnings of society.