it’s a modern phenomenon really as urban centres get bigger. Now there’s two distinct categorisations of a city - the city itself and the wider city region. In this country often referred to as the metropolitan borough.
Greater Manchester is the perfect example. A lot of people say they live in Manchester when they actually don’t. Bolton, Stockport, Oldham, there’s loads of cities around the city that benefit from being part of the wider metropolis.
every major US city is like this too.
It plays havoc with population counts though. In some lists, Liverpool is bigger than Manchester for population. It depends where you draw the boundaries. Liverpool city region actually takes in The Wirral, Sefton, Knowsley, St Helens and Halton and is actually fucking massive (and growing - it keeps swallowing small towns on the perimeter. Where I live once fell under West Lancashire. Not any more)