His entire way of living. The old English style manor that has been in the family for generations. The people that he must associate himself as Bruce Wayne are also part of that same world.
The old English butler that has been in the family for decades. Those are just to name a few. Those are all ways of living that I someone of any other background wouldn't care emulating to the T if they weren't of a WASP background and if they weren't and decided to do so then they are following something that was created by a specific type of European wealth that is mainly attributed to a the Anglo Saxon group I'm referring to.
1: Pretty much all of those things could easily be embodied by a non-white person. Rich black families can still live in mansions, even old English style mansions. It might have been in the family for three generations instead of eight, but what difference does that make? A a black or latino man who's the majority shareholder in a major corporation is still going to rub shoulders with rich white dudes constantly, because that's the nature of the corporate world, so I don't see why Bruce Wayne's social calendar would be any different. And being white isn't a prerequisite of hiring and English butler. Having a huge house you don't have time to take care of yourself and finding a guy from the UK who's suited for the job is really the only prerequisite required.
2: With the exception of Alfred, which, again, could be there regardless of the Wayne Family's race, those things aren't the core things that define Batman. Batman isn't defined by how long his family has lived in Wayne Manor, and he's only marginally defined by the social circles he travels in out of mask (and would very likely still travel in regardless of his race). He's defined by his pain, and by his conviction, and by the boundless compassion that hides behind all of that. Everything else is just details.
Rich white dude. Scion of a old rich white family. Lives in an old rich manor.
What does Bruce have to do to more embody rich old whiteness? Wear a waist sweater?
But he only embodies rich old whiteness because of those traits. If he didn't have those traits, nothing else about who he is or what he's about would change. If he was a rich non-white dude, the scion of a young rich non-white family who lives in an old rich manor, he would still otherwise be Batman. If nothing else about the character would change, besides maybe giving him some new conflicts and ethical dilemmas that might actually add a new layer of richness to the character, then why are those three aspects so vital?
I'm sorry, but what makes blank have to be blank (someone said Wonder Woman, another said Black Panther), but not Bruce Wayne? Is white privileged old money background not an important part of his backstory? Why doesn't that count?
Is it not part of his image? Sure looks like it is to me.
This seems arbitrary to me.
Because:
1: Bruce isn't as strongly defined by his cultural background as Black panther is. He just isn't. He doesn't wear the holy vestments of his proud Scottish ancestors, he doesn't regularly and openly practice the sacred cultural rights of his people, he doesn't fight against his people's ancestral enemies in a modern world. It's there, but it's not there as much as it is for someone like T'Challa. The two aren't equal in that regard. A black Batman would be largely the same as a white Batman. The same can't be said for a white T'Challa.
2: Black Panther's cultural heritage is a much rarer thing in popular media than Batman's. It need's to be preserved. There are plenty of white Anglos in popular media, if we lose a few we'd be just fine.