Yes, but gender's nature as a "performance" can be easy to misunderstand. People often conflate gender being a performance with someone needing to perform a gender to be the gender; that by performing a gender that is who you really are. That gender only exists as a social phenomenon performed for the sake of others.

It's more that we're all actors who are given roles to perform, and while most people are content playing their given role, some people are not. Even outside of social pressure, people have their preferences, with most people suffering a mental consequence if they can't perform their preferred role. Even though nobody is born embodying a gender, they can not be assigned any gender other people choose.

It's also important to note that while gender is malleable and can very greatly across time and space, there are poorly understood degrees of freedom for gender. We are so, so far from knowing why people have gender preferences or what defines them. It probably isn't the case that there's some universal essence of gender, with us instead having underlying psychological patterns that express themselves within the social context. We might not be able to choose all genders equally, but we can choose what fits best and redefine gender for ourselves.