As far as I can determine, it’s about the intersection of consent and context.
A random, white civilian wearing a Purple Heart they bought on eBay is problematic because they didn’t earn it in the manner in which it is awarded; they are making a false claim and/or removing the meaning from the symbol.
That same person claiming to be a mambo in Vodoun is problematic for the same reasons, unless they actually went through the proper initiations with someone of the lineage (it’s not strictly geographical or racial but you have to be taught by someone qualified, who will make sure you are getting the right context and so on).
That same person teaching a Bible study class, or practicing yoga, or playing taiko? All fine because those practices were not meant to be exclusive. Consent (or active encouragement) is given.
When I was trying to figure out my gender identity I came across the term “two-spirit” and immediately wanted to use it, because I had actually written about my feelings with very similar phrasing in the past. And it’s even a modern term (from… 1998 I think?), though it references an identity that various Native American cultures have recognized for ages. But they explicitly withhold permission for people not of those cultures to use it. I sometimes find that slightly irksome, but I respect their wishes.
Things can also get a bit weird when the culture in question is being recreated – sometimes people claim to speak for them while making racist claims themselves (e.g. Neo-Nazis infecting Heathenry; a Celtic pagan teacher requiring potential students to submit DNA tests; Black people telling an Egyptian reconstructionist temple that Egyptian religion is not for white people or Asians, etc.) My spouse is a Lokean who won’t wear runic jewelery because she doesn’t want people thinking she’s a white supremacist.
