Nor "Frenchie" while we're on the topic. It sounds really weird. It's also commonly used to refer to a french bulldog !

I would think of using 'Frenchie' to refer to a person as being affectionate banter. Like 'Yank' for Americans or 'Canuck' for Canadians. It's not incorrect, but would be inappropriate outside of an informal context.

French people have 'rosbif' to refer to the English and Australians have 'pom' or 'pommie'. You wouldn't call the prime minister that at a diplomatic event, but it's not offensive to call your friends that.