Well I don't think Charles' comment was racist - I think he was trying to engage the woman in conversation.
I think she knew perfectly well what he was trying to establish, but she chose to reply in that facetious way.
I imagine he was expecting her to say that although she lives in Manchester where she was born, her family came from, say, a Caribbean island or an African country.
This would have been an opening for Charles to maybe say that it's a beautiful island or country which he had visited a number of times, and thereby get the conversation going.
Think about it - the Royals have all these people to meet and converse with day after day. It must be SO difficult to think of something different to say when faced with probably very nervous people.
Of course there will always be those who seem to enjoy trawling through the internet or newspapers looking for some 'racist' comment or other to be outraged at.