Rachel Dolezal is good point. Ethnicity, like Sex/gender, isn’t solely biological either is it. If you were raised as and look and feel White European then find out via dna testing you’re over a quarter Native American would you try to acculturate your way in to those communities?
There's two separate issues which shouldn't be conflated, in my opinion. The purely physical issue, as it pertains to the transgender debate (but not the trans-racial analogue), namely that men are bigger, stronger, and have penises with which they rape women (the fact that they account for 90% of violent criminals in prison and 98% of sex offenders just is an indisputable fact). That's the argument that applies to the sheer, unadulterated stupidity (not to mention evil intent) of putting Karen White in a women's prison. Then there's the social issue of male socialisation and privilege. That's the argument which shows the absurdity of giving a women in business award to Pippa Bunce.
Only the latter argument, about socialisation, privilege, the weight of history, applies to the Rachel Dolzeal's of this world. If you've grown up thinking you're white, benefiting from white privilege, not being discriminated against, and you do an ancestry DNA test and find that you're 1/4 Native American* then apply for a scholarship intended to get kids from underprivileged backgrounds on reservations into college - you are a total twat for doing so.
(*FWIW my DS is 1/8 BAME by ancestry - but looks white and due to family background will definitely grow up with white middle-class privilege however you cut it. No way would I be encouraging him to take advantage of affirmative action programmes, because this would be utterly shitty of both of us.)