Yeah, but, to nitpick - just saying "Chinese" (language) can be assumed to mean Mandarin, at least from the point of view of the "national" language (Mainland government is pushing Mandarin education, and so are Taiwan and Singapore, etc). The same could have been said about the written language, except the traditional / simplified divide.
For cuisine, on the other hand, the generalization is much more
lossy. Dishes from one part of Greater China can have very little similarity to dishes from a region on the other side of the country.