China has 8 main dialects with multiple sub dialects and you are correct, many cannot understand one another, good example you supplied, Mandarin speakers and Cantonese speakers cannot understand each other. However they all share the same writing system. However these days in China, most speak Mandarin, it is the national language, but they also speak there local dialect as well.
When I was stationed on Okinawa, the native tongue spoken by elders was referred to as hogun, and it wasn't a dialect of Japanese but a separate language as I was informed. I am no expert.
It could be a separate language in practice, whether or not is officially recognized. I think most people assume that "dialect" is an objective distinction that refers to variations within a single language, something akin to the term "accent" but more extreme.
It's not just a dialect but almost a completely different language in some areas. For example Kagoshima ben is incomprehensible to those from over the river. Some new Kindergarten kids join speaking just a dialect as most have spent a considerable time with grandparents at home. In my area Watakushi desu "Its me" becomes "Oi batenga". The very words my brain surgeon friend would use with me but he changed into standard Japanese when talking to patients as do most business people.
Yes it totally different. I had an Okinawan friend that used to lapse into it after few drinks.
