Gender seems to have psychological, social, and biological components. Julia Serano covers this territory fairly well in Whipping Girl, esp. chapter 6 "Intrinsic Inclinations":

... [T]he fact that gender expression is so highly regulated in our society has led many to argue that femininity and masculinity are merely social constructs (i.e., they do not occur naturally, but rather are inventions or artifacts of human culture). According to this social constructionist model, boys are socialized to become masculine and girls feminine; we learn to produce these gender expressions via a combination of positive and negative reinforcement, and through imitation, practice, and performance. Social constructionists point to the fact that the words “femininity” and “masculinity” do not merely describe human behavior, but represent ideals that all people are encouraged to meet. To demonstrate this, they focus much of their attention on socially influenced manifestations of gender expression (often called gender roles), which include feminine and masculine differences in speech patterns and word choice, mannerisms, roles in relationships, styles of dress, aesthetic preferences, interests, occupations, and so on. Social constructionists also argue that the fact that these gender roles can vary over time, and from culture to culture, is indicative of their constructed nature.

On the other side of this debate are gender essentialists, who believe that those born male are simply preprogrammed to act masculine, and those born female are preprogrammed to act feminine. Evidence to support their case includes the predominance of femininity in women and masculinity in men, in our culture and other cultures; the fact that girls tend to behave in a girlish manner and boys in a boyish manner from a very early age; that even in prehistoric humans, women and men seemed to perform different sets of tasks; and that species other than humans also show signs of gender dimorphic behavior. Among gender essentialists, it’s generally assumed that genetic (and subsequent anatomical and hormonal) differences between females and males are the ultimate source for these behavioral differences. Despite their insistence, such direct links between specific genes and specific gendered behaviors in humans continue to remain elusive.

As someone who both is a geneticist and has experienced firsthand the very different ways in which women and men are treated and valued in our society, I believe that both social constructionists and gender essentialists are wrong (or at least they are both only partially right). The fatal flaw of the gender essentialist argument is the obvious fact that not all men are masculine and not all women are feminine. There are exceptional gender expressions: There are masculine women, feminine men, and people of both sexes who express combinations of femininity and masculinity. People who have exceptional gender expressions (like those with exceptional subconscious sexes and sexual orientations) exist in virtually all cultures and throughout history, which suggests that they represent a natural phenomenon. Gender essentialists often try to dismiss such exceptions as anomalies, the result of biological errors or developmental defects. However, exceptional gender expressions, subconscious sexes, and sexual orientations all occur at frequencies that are several orders of magnitude higher than one would expect if they represented genetic “mistakes.”2 Further, the fact that we actively encourage boys to be masculine, and ostracize and ridicule them if they act feminine (and vice versa for girls), strongly suggests that were it not for socialization, there would be even more exceptional gender expression than there is now.

Unfortunately, a strict social constructionist model does not easily account for exceptional gender expression either. Many girls who are masculine and boys who are feminine show signs of such behavior at a very early age (often before such children have been fully socialized with regard to gender norms), and generally continue to express such behavior into adulthood (despite the extreme amount of societal pressure that we place on individuals to reproduce gender expression appropriate for their assigned sex). This strongly suggests that certain expressions of femininity and masculinity represent deep, subconscious inclinations in a manner similar to those of sexual orientation and subconscious sex. (I use the word “inclination” here as a catchall phrase to describe any persistent desire, affinity, or urge that predisposes us toward particular gender and sexual expressions and experiences.) While I believe that such inclinations are likely to be hardwired into our brains (as they exist on a subconscious level and often remain constant throughout our lives), I hesitate to define them as purely biological phenomena, as social factors clearly play a strong role in how each individual interprets these inclinations. In fact, in most cases it is impossible to distinguish our inclinations from our socialization, since they both typically point us in the same direction. Generally, we only ever notice our inclinations when they are exceptional—when they deviate from both biological and social norms.

Further evidence that gender inclinations represent naturally occurring phenomena can be found in other species. If one looks across a wide spectrum of mammals and birds (whose gender and sexual expressions are presumably not shaped by social constructs to the extent that ours are), one generally finds certain behaviors and affinities that seem to predominate in one sex, but which also occur at lower but substantial frequencies in the other sex as well.3 Thus, any model that attempts to explain human gender expression, sexual orientation, and subconscious sex must take into account the fact that both typical and exceptional forms of these inclinations occur naturally (i.e., without social influence) to varying degrees.

In order to reconcile this issue, I would like to put forward what I call an intrinsic inclination model to explain human gender and sexual variation. Here are the basic tenets of this model:

1. Subconscious sex, gender expression, and sexual orientation represent separate gender inclinations that are determined largely independently of one another. (This model does not preclude the possibility that these three inclinations may themselves be composed of multiple, separable inclinations, or that additional gender inclinations may exist as well.)
2. These gender inclinations are, to some extent, intrinsic to our persons, as they occur on a deep, subconscious level and generally remain intact despite social influences and conscious attempts by individuals to purge, repress, or ignore them.
3. Because no single genetic, anatomical, hormonal, environmental, or psychological factor has ever been found to directly cause any of these gender inclinations, we can assume that they are quantitative traits (i.e., multiple factors determine them through complex interactions). As a result, rather than producing discrete classes (such as feminine and masculine; attraction to women or men), each inclination shows a continuous range of possible outcomes.
4. Each of these inclinations roughly correlates with physical sex, resulting in a bimodal distribution pattern (i.e., two overlapping bell curves) similar to that seen for other gender differences, such as height.4 While it may be true that, on average, men are taller than women, such a statement becomes virtually meaningless when one examines individual people, as any given woman may be taller than any given man. Most people have heights that are relatively close to the average, but others fall in outlying areas of the range (for instance, some women are 6 feet 2 inches and some men are 5 feet 4 inches). Similarly, while women on average are more feminine than men, some women are more masculine than certain men, and some men more feminine than certain women. Because these inclinations appear to have multiple inputs and show a continuous range of outcomes, it is incorrect to assume that those with exceptional sexual orientations, subconscious sexes, or gender expressions represent developmental, biological, or environmental “errors”; rather, they are naturally occurring examples of human variation.

In terms of what you have asked me, I believe gender identity is biological in the sense that your subconscious sex (as Julia Serano would call it) is not something you can choose or that can be altered by social influence. I believe this to be grounded empirically, in the fact that conversion therapy does not successfully treat gender dysphoria while transitioning does. The conservative medical establishment would not back transitioning otherwise, if conversion therapy worked, our cis-normative society would absolutely endorse it as the main treatment for gender dysphoria. There is of course additional evidence in the MRI scans and the autopsies of trans brains which found trans women had structures in the brain like cis women, the brain-sex mosaic that was discovered and so on.

What this means for trans and non-binary folks is that our experiences are not the result of social contagion, delusions, or imagination, but instead a result of natural variation and our biology, even if the way that implicit gender identity manifests in our personal and social lives is clearly shaped by cultural influences.

It also means that conversion therapy, as established empirically, cannot be effective because it cannot change subconscious sex or the causes of gender dysphoria.

If gender is just a social contruct, why do trans people want to change their gender?