I'd say we certainly have a baseline that has been very successful for most people. The issue is that it might not work for all people, and there are some that seem heavily invested in the presumption of organs providing identity instead of the reality being a bit more complicated, and that it merely seems like the organs are providing the identity. Kind of like gravitation in Newtonian Physics and Relativistic Physics.
I think sex might simply be your sexual identity and gender is how society allows you to manage it.
"Identity" is an inherently social question; inanimate objects and non-social organisms do not have identities. We know, then, that any sense of "identity" must be on some level socially constructed. Those who are not versed in the social sciences often think that "constructed" means "fake", but that's not really what social construction means. Rather, we take the scaffolding of the observable universe and construct our stories and narratives around it. We build social constructs like gender around the physical traits that denote sex, race around certain favored phenotypes, intelligence around certain cognitive functions, etc. But whatever the objectively observable facts that might underlie some of those concepts, we cannot resist the seem to resist the urge to embellish, categorise, reinvent, narrativize, and anthropomorphize the natural world. The very language of science is infected with this plague, with our talk of "laws" and "constants" and "taxonomy"; social terms applied to natural phenomena that neither think nor feel anything about their own nature or how they "ought" to be organized into comfortingly simple and non-overlapping groups.
This is not something I entirely agree with as a statement.
Many things, indeed anything capable of rendering behavior as a result of parsing some context via some arbitrarily configured system are going to have "identities" within them operating as drivers.
The identity is literally the state of the system that has been configured in such a way as to produce that "truth" of the system.
Under this concept of identity, even a CPU has a number of identities, namely identity statements for each of its instructions, formed of the microcode for those instructions.
I pose that the identities humans have are fundamentally similar in the form of some arbitrary configuration of neurons which produce "systemic facts" fundamental to the operation of whatever-it-is.
The configuration of my brain that determines why I feel "feminine" when I do certain things is a part of my "identity", and because it relates to gender, is part of my "gender identity" and this is true even if I am the only human or other entity on the planet.