Really? How is this any different from working class people or women generally joining the military? Does that make the the Army, Marines, National Guard, etc. "their" organisation? The upper echelons of the military and police in the US and UK are dominated by white, economically privileged, straight men from a Judeo-Christian cultural tradition and those who suffer most from their violence aren't. Still, working class folks, women, Muslims, people of colour, Lesbians and gay men join both institutions, for all sorts of reasons. Maybe they think their best chance of surviving in an unjust society is carving a niche within one of it's unjust institutions - being a "good" woman, Black guy, Muslim, etc., but knowing being part of "us" and not one of "them" is conditional. And, they'll probably be regarded as traitors by others who didn't choose as they did.
Structural oppression IS based on identity though. Those at the top of the social/economic/political pile determine respective value and status of those below them on the basis of identities, often pitting them against each other. Otherwise, you wouldn't get wealthy Black people subject to racism, female public figures subject to threats of sexual violence, passive-aggressive slating of politicians with working class roots for their accent and manners, etc.