He is sometimes, yes. But he also fights against wealthy criminals and seems liked by the upstanding poor. Does the poor person getting harassed by a gang or mugged on the street think of Batman as "one of us?"
Historically, that's very different from being working-class, for instance.
Sure, different from working class, though siding with the ruling class meant the serving class knowing their place. Alfred's background now is that he was an actor for a while and was in the army, and has relatievs who are actors. He has all the polish of a person living amongst the elite, but he brings a little more color to the table than that.
I'm not sure it's fair to view Dick in the same way as the later kids as far as author intent is concerned, since his editorial history is so different and his social class didn't matter (much in the same way character psychology didn't matter) at the time he was created.
Yet his class wasn't changed when things started being played differently. It was made more part of his character. Even if his creators weren't originally planning to deal with the psychology of social class they still decided that their millionaire should take in an orphan from the circus rather than a kid from his background--I think it did matter. They would knew what kind of class a Depression-era traveling circus implied.
I don't really have my own argument for what it all means. I feel like there's a lot of different things at play along with the backgrounds. Thinking of the main team I get (hope I'm not making mistakes here):
Damian--European aristocratic Bruce--super rich, old money, rare background. Alfred--Family tradition of domestic service, supported himself as an actor and in the army. Tim--Upper middle class. Wealthy enough for prep school etc., but not in Bruce's league. Gordon--A cop, working often as a partner with Batman. Barbara--Same background as Gordon. Helena--Money, but mafia money. Dick--family of headliners in a small circus, so brought up with working class people. Steph--Lower middle class? Her mother's a nurse? Her dad's been in jail etc. Jason--Homeless, underclass. Cass--I suppose fairly wealthy beginnings, but bizarrely brought up, and then homeless when taken in.
For status I use "immoral" for any problems raised in that area.
Damian--Healthy, immoral, skilled Bruce--Fake death, moral, skilled Alfred--Healthy, moral, skilled Tim--Healthy, moral (immoral in possible future, flirts with immorality, "crossing the line"), skilled Gordon--healthy, moral, skilled Barbara--disabled, moral, skilled Dick--healthy, moral, skilled Steph--died/resurrected, moral, unskilled Jason--died/resurrected, mental issues/immoral, skilled Cass--healthy, moral (turned evil temporarily), skilled, dropped