I'm not convinced this 'Manchester = united' when speaking to people overseas is true at all.
It's more of a generational thing in my experience.
I was born and raised in Manchester, but have been lucky enough to find a career that has allowed me to live and work around the World. Anytime I've chatted about where I'm from and mention Manchester, the first question tends to be "City or united?". Certainly with people 30 or under.
Mrs Munson is American, and we live on the East coast. All of her nieces and nephews and their friends (all in their late teens/early 20's) know as much about City, if not more, than united.
I work for a major network's streaming platform over here. I get paid to study trends, views, etc. and subsequently advise on content production. I know for a fact that the Manchester = united tide has shifted 'massively' over the last few years and will continue to do so.