Being in London reminded me of something I've been considering lately, namely that London, along with New York, is that rare city that is at once representative of its wider country but also completely unlike the rest of the country.
To explain: both London and New York are the centers of finance, culture and business for their respective countries, which means that foreigners associate the UK and the US, respectively, with them. For a lot of non-Americans, New York is considered the most obvious expression of America, and for non-Brits London holds the same position as the archetypal British city.
But at the same time, both countries are also just diverse enough that nowhere else is like New York or London, to the point that residents of some areas define themselves in opposition to these cities. Examples are how Southerners or West Coast people (or even folks from the Midwest) hold up New York as The Enemy, an example not to be followed at any cost for its poverty and decadence. With London, the cultural resistance comes from the North, where anyone south of the Midlands is regarded as a "southern nancy", and the Home Counties, which define themselves in suburban opposition to the metropolis. For some Americans, New York is pretty un-American, while for some Brits London isn't very British at all.
I find it an interesting idea, because I have trouble thinking of another city that occupies the same place for both foreigners and locals. The closest is Paris, which occupies that same space in foreigners' minds of being so comprehensively French, even if the rest of the country can be quite different. Yet I'd have trouble imagining that folks from other regions in France consider themselves to be more truly French than Parisians.
Even countries like Italy and Spain don't seem to have this dynamic. In both of those cases, you could argue that the capital (Rome and Madrid) has a key rival in another part of the country (Milan and Barcelona), and that both are equally representative of their respective countries' essences. As different as the various regions of Italy are, I don't think I've ever encountered any Italian who would argue that Rome isn't very Italian. By contrast, Spain is so diverse and linguistically fragmented that many in Barcelona, or Catalonia more generally, define themselves and their city as less Spanish.
Understanding of a place also plays into this. India is quite a diverse country, along ethnic, linguistic and religious lines, to the point that it's hard (for me at least) to consider one city, say New Delhi, as more quintessentially Indian than anywhere else. Same with China - I don't know how people in Sichuan or other regions sees Beijing, but given that it's where the seat of power has resided for centuries, I would assume it's considered very Chinese even by locals.
But the overall dynamic of London and New York remains really interesting to me, in part because it's a truism that both are more similar to each other than to other big cities in their regions. This isn't to say that New York is a particularly European city (it really isn't), or that London is at all American, but it is true that they have a lot of similarities that make comparisons between them meaningful. It feels nonsensical to compare Chicago or LA with London, just as it feels weird to compare Rome or Amsterdam with New York (to say nothing of Edinburgh or Birmingham).
But comparing London with New York does make more sense - and from that comes their status of being both representative of, and unique within, their home countries.