The distinction between Caitlyn Jenner and Rachel Dolezal can unfortunately be difficult to grasp without some background in sociology, feminist theory, queer theory, and a little psychology. I’ll attempt to provide this context below.
(Disclaimer: I am Malaysian-Chinese, and my knowledge has been learned from other Black, trans, and other people of colour. Anyone with that lived experience who wants to correct or add anything are welcome to do so.)
Psychology - Children generally develop a stable sense of their gender identity by age 4, at which point they begin to learn gender norms. Most transgender folk innately know their gender identity at this very young age, or at some other point in childhood, but unfortunately do not feel free to transition or express their true, felt identity. Rachel Dolezal, however, was raised entirely as a white individual through to adulthood. She has no lived experience as a black woman. At some point in her adulthood, she decided to construct a faux-black identity based entirely on physical appearance. What makes her ‘blackness’ fake and illegitimate is that gender is a much more open, fluid, and accommodating construct than is race. Outside of the gender binary, people identify as androgyne, two-spirit/bigender, multigender, agender, nonbinary and more. Please feel free to look them up.
Race is a concept that early/proto-anthropologists and scientists formally created in the 17th century. 1735 - Carolus Linnaeus categorises the human species into racial categories defined by continent and skin colour, later attaching stereotypical 'temperaments’. 1775 - Johann Friedrich Blumenbach proposes racial categories based on physical features, but does not assign hierarchies. Prior to the 17th century, ancient civilisations (e.g. Ancient Greece) had what we would call precursors to race, and were almost always tied to a hierarchy of higher and lower, better or worse. Historically and currently, the concept of race has always been inextricably tied to racial supremacy, oppression, and discrimination. The dominant understanding of race in North America is this early European model that categorises the world’s people, mixed with the US 'one-drop rule’ based on blood/ancestry. In this framework, you are either one race, or biracial/mixed. Many mixed people choose which side to identify, and this most often is tied to their appearance.
Point being, Rachel Dolezal cannot be compared to biracial/mixed people, as BOTH her parents are of white, European descent. She is literally as white as can be. The defining characteristics of race are physical, based on blood, family, and ancestry. A black child adopted by white parents may become culturally white, but their race does not change. Race, unlike gender, is immutable at the individual level. There is zero ancestral basis for her to claim to be black. Also keep in mind that she’s inconsistent with her racial identification. She sometimes claims to be fully African-American/Black, or biracial (White and Black), or mixed (White, Black, Native American). These are contradictory positions, and she cannot make up her mind precisely because these are fake identities she constructed for herself.
Feminist theory (Judith Butler) - Gender is something that we express, present, and perform through our actions, appearance, and countless little things in life. Note that this is a pretty radical and somewhat dated (Gender Performativity, 1990) view, though it’s quite influential. Current understandings of gender are a meld of 'stable innate self-construct’ and 'fluid performance’. It might seem contradictory, but my way of understanding it is by visualising gender as a half-melted sweet, simultaneously solid and liquid. Gender has both an innate component relating to feelings about their body, and a socially constructed, performed component. The innate component is what determines 'true gender’. It is self-identified.
Sociology - What Rachel Dolezal does is cultural appropriation to the absolute extreme. She claims an entire culture and racial identity for herself that she has fabricated. In contrast to 'actual’ Trans-racial and Trans-ethnic people (who are people of one race/ethnicity that were adopted by parents of another race/ethnicity), what she does is study and imitate a few markers of blackness (hair, darkened skin) and attempt to pass herself off as being of that race. It’s comparable to white anime fans (Weaboos) who suddenly 'realise’ that they’re trans-racially Japanese, based on shallow, stereotypical and limited understandings of Japanese culture. I’m quite surprised that it worked, because in hindsight she looks incredibly fake and nothing like any of the lightskinned black people or mixed-race people I’ve ever seen. The fake tanner seems obvious.
What additionally separates Rachel from actual Black people is that she retains her white privilege. She can wash off the tanner, grow out her natural (straight, thin) blonde hair and reclaim her white identity (this is the particularly egregious part of cultural appropriation). Actual black people can’t do that, and that’s why you hear non-white people joking about how they wish they could claim to be transracially white to escape racial discrimination.
Queer & Trans Activism - This new wave of white people trying to 'become’ other races are insulting both transracial/transethnic adoptees, and co-opting trans terminology in an offensive manner. The whole “If Caitlyn can be a woman then Rachel can be black” response waters down and de-legitimizes trans identities/experience by assuming trans women are just 'pretending’ and we should smile and play along. This is patronising, not accepting. For the reasons I’ve discussed above, race and trans identities may both be social constructs, but are not equivalent. Their historical context, definition, criterion, and usage are significantly different.
I hope that this helps someone understand the situation better.