Yeah, as a kid who was raised on a council estate I do wonder if this whole class thing is just about access to money and education. I got the latter from the state, and haven't seriously suffered from lack of the former despite my best efforts. Money might have helped, or hindered. I'll never know because I always had just enough. Education, on the other hand, should be a universal right (and it's a priority of the United Nations, specifically UNESCO.)
So it's a richly amusing to me that the two characters are filmed in the same location.
Both Rose and Clara are "secret life" dramas. They plot the integration of comics and other niche media into everyday life. Both are creations of writers who have been intensely affected by the sexual revolution. Russell Davies and Steven Moffat create strong characters, but with a more-or-less equal gender emphasis. Both of them satirise and utilise the tendency towards normalisation. A young girl saves the world in one episode but everybody has forgotten it in the next. Yes, that is satire, and one of the greatest gifts of New Doctor Who. Clara differs from Rose only in the fact that she already knows she's all that.
Well, it's a bit of a weird one to be specific - for Rose, they actually went to a genuinely rough council estate in London for a lot of exterior wide stuff - and there's amazing production stories of people walking on set in the middle of a take of Tennant's regeneration stagger to the TARDIS shouting "Fucking BBC! You're all poofs!" and the like, or kids pissing on the TARDIS between takes, a member of the production getting punched for trying to stop a man walking into shot, all that. So, Rose's background of being 'working class' was grounded in that real location.
That was then supplemented with a slightly posher area in Cardiff that was a good visual match, and that was used for close-up stuff, the immediate exterior of the Tyler's flat etc - because obviously they could only afford a few days in London per year. Most notably the staircase we see Rose and the Doctor go down during 'Rose' is used for it - a staircase we see Clara ascend again in a very similar shot in Time of the Doctor, and that's the biggest visual clue it's the same location in the show itself.
So, it's not entirely the same location in that the actual 'feel' of Rose's estate was really provided by a different place.
On the subject of class and Doctor Who, I don't disagree that fundamentally the characters are the same, but class is a very real thing (even if it is just a social construct), and it is true that old Who was a fundamentally middle class show by its very nature, by the people it cast, and the accents they spoke in. RTD openly said he made a concerted effort to reverse that with the revival, and that's really how you got Eccleston and Rose, the character, played by Piper speaking in her accent that isn't her own (cos she's posh for real). The Radio Times ran a fascinating article about the BARB and BBC's research into the social groups - ABC1 and all that - and talked a bit about how, under Moffat, Doctor Who is watched by less people in the 'Working Class' categories and more people in the 'Middle Class' categories, probably because Amy lived in a massive house in a small country town, Clara is a seemingly well-off teacher, etc; it does trickle down and make these people more or less relatable to a lot of the audience. It's a rare show that somehow speaks to both (Only Fools and Horses was notorious at doing this, somehow.) It's not a good or bad thing, it's just an interesting thing to look at, the way characters and tone can impact who watches like that.
In line with that, I imagine Capaldi wouldn't have made the comments about wanting to see a working class salt-of-the-earth companion next if he didn't know Moffat was going in that direction, so maybe there's going to be a push to win those people back.