A social system that constantly organises people by gender.
Assuming that here you are using 'gender' to mean not sex but societal expectations, this is precisely where the problem I have with 'gender identity' originates. Society, or to more more accurate, two parts of society (the 'progressive left' and the 'conservative right') give the concept of gender far more significance than it warrants.
Our personalities are all made up of many factors, and centring one of these excessively is unhealthy. In the case of 'gender identity', it can lead to various unhealthy outcome, including estrangement from family, the disruption of endocrine system, and cosmetic surgery; these can in some cases cause further psychiatric and medical problems, and are leading to societal division and mental health problems for some of those around the trans-identifying person.
The 'gender identity' part of personality is far from the only one that can be problematic when it becomes too central. It perhaps explains, though, the noticeable correlation between autism and trans identification. When it is promoted consistently and ubiquitously, it can outcompete other interests and models for understanding one's place in society, and is attractive to those who feel that they are misfits. Once entrenched as the defining issue of a person's identity, anything pushing for another worldview is easily framed as 'hate'.