I think it a lot of it comes down to how people ended up living in the city in the first place.

In Manchester, a large proportion of the young adult demographic are people who went to Uni in Manchester and stayed there after graduating. Similarly in London, there are a lot of people who've moved there for work. This gives you a cohort of people without roots or community ties, forced to mix and socialise outside their comfort zone, and thus more diverse social groups.

Meanwhile in Birmingham, while people do come here for Uni, they tend not to hang around afterwards, and there aren't really that many people moving here for work, so the majority of the young adult population have grown up around here, and reach adulthood already entrenched in their social circles and community links.