What's the deal with the Black Country? Is it just ex-industrial suburbs of Birmingham a la the towns around Pittsburgh?
Following up on that, to what extent do people there identify with Birmingham? I.e. would a person from Wolverhampton describe themselves as from Birmingham to someone from elsewhere?
The answers to your two questions are related: the towns of the Black Country are very much not seen as suburbs of Birmingham, but rather have a fierce local, independent identity. I imagine many people from the Black Country would get rather offended if you called them Brummies!*
(As a side note, your comparison to America more broadly highlights an issue of scale in comparing between the two countries: Wolverhampton is a one hour drive from the centre of Birmingham, which, regardless of local identities, is generally too far for somewhere to be considered a suburb of a city in this country, with the possible exception of Greater London.)
*Although as an anecdote, my grandfather, who lived as a child in both Smethwick, which is right on the border of Birmingham and the Black Country, and in Birmingham itself, unsurprisingly did describe himself as being from both places.
