I think it's even funnier to see how many Chinese people - even those Chinese people who are fluent in English - take the folksy word 方言 and try and shoehorn that into the precise linguistic term "dialect". For whatever reason Chinese the world over seem to believe in literal 1:1 translations of Chinese -> English terms, when they really don't mean the same thing. It makes it very hard to talk to them.

We all learn classical Chinese at school, and that must be a completely different language.

Of course it's a different language. It's not completely different. Languages can be related to each other. Old English is a different language than English. If English was written in Logograms instead of a phonetic alphabet, many of us could probably read Beowulf the same many of you can read the works of 老子.

Again, if I had the Chinese mindset, there would be a language called "Indo European" which would have various dialects like "Persian" and "Russian" and "Icelandic". There would also be "Standard Indo-European" - aka, English. It makes no sense.