MotherofDinosaurs I might not get the explanation 100% correct but this is my take on it.
I am also mixed race, FWIW.
The thing is no-one 'looks' like they are from Manchester - they might sound like it but even those that do sound like it might not actually be Mancunian or indeed even be English. I too am always happy to talk about my origins. I too don't look like a native English Brit but then again some of my European relatives with white skin, blue eyes and blonde hair are not English though they might speak fluent English with an English accent.
The problem arises when the so-called leader of the common-wealth, a man whose father thinks nothing of making nigh on racist remarks deems it acceptable to casually drop into conversation that someone that's told them that they are from Manchester decides to retort: "Well you don't look like you're from Manchester". Considering this person is a POC makes it a very dog-whistle racist remark. It makes it even more pertinent that this is a white highly privileged person making the remark in the first place.
My exH was Dutch, blonde hair and blue eyes but spoke English with a heavy accent. Who was to know where he was from - he could arguably pass for any European country. I am racially ambiguous so could arguably be from anywhere.
It's about looking like you 'belong'. The problem is that of course she doesn't look Mancunian but she is. Similarly to those that have said on this thread that they are from elsewhere but consider themselves to be xyz. In his position he should know better than to make such comments. He should just smile and nod along without making that woman feel like an alien in a place she considers home.
I personally think the Commonwealth is a huge pile of wank. It'd have been more helpful had the Brits not conquered and raped and pilfered from two of my nations but here we are. He needs to be held accountable on two counts: 1) He is supposed to be the next monarch of this country and 2) He should know better how to make polite conversation and chivvy things along without offending brown people.