I guess I'll provide the obvious follow-up to the Black Country Question: Is it really still the case?
Yes, there are plenty of examples worldwide of adjacent economic cores developing differently, maintaining differences, and ending up in different places with different cultures. And those cultures like to persist even when the resident populations shift and times change. There's no denying that.
However, in the modern age where everyone wants to be in or near a city, cause that's where the good paying jobs are, areas that might have once been distinct end up in an unholy mess of urbanization. The most obvious example that springs to my mind is San Francisco and Oakland, but there are many, many more worldwide.
It's not that hard to get past census data to examine this. As a percentage of the resident working age population, you still see larger shares commuters coming in the from the areas to the north, east, and south of Birmingham - the Black County does provide for some of her own. But in terms of raw people, Sandwell, Walsall, and Dudley are numbers 2 through 4 in terms of thousands of commuters to Birmingham. Sandwell is only surpassed slightly by Solihull. Conversely, as a sign of the urban integration, people from Birmingham go in the other direction for work. Solihull has the number one spot by far for Birmingham -> somewhere else, but Sandwell and Walsall are 2 and 3.
If you look at the official
Travel to Work Areas you will note that there is a Birmingham TTWA which extends from the city north to Tamworth and south to Redditch and Bromsgrove and also includes urban Solihull (Solihull proper plus Chelmsley Wood and Castle Bromwich). None of this is very surprising.
There is also a Dudley TTWA which includes pretty much all of that borough together with a few adjoining areas in Staffordshire and Worcestershire together with most of Sandwell: Oldbury, West Brom, Tipton and Wednesbury. Then there is a rather extensive Wolverhampton & Walsall TTWA which includes pretty much all of the former and most of the latter, but also extends all the way to Lichfield and Cannock as well as west towards Telford (but not getting that far).
There are however bits of the "Black Country boroughs" which are in the Birmingham TTWA. Some of these are I think really spillover of genuine Birmingham suburbia and not really part of the Black Country in spite of the administrative boundary: Pheasey, Streetly, Great Barr. Then there's Smethwick, which I think is usually regarded as Black Country but more associated with Birmingham than the rest of it, and then some of the Warley area to the south of it which was also part of the old Smethwick borough.
However that link does also show you some "alternative TTWAs" using some subsets of occupations, and it does in fact turn out that for some of the higher status ones (e.g. the high qualifications one) the Birmingham, Dudley and Wolverhampton/Walsall TTWAs do all merge into one (but Coventry stays separate). And some weird things happen for some of the others too.