This old Language Log post has a good demonstration of how Cantonese relates to Mandarin, by positing a series "German and English are just dialects" examples.
languagelog.ldc.upenn.edu/nll/?p=6654
languagelog.ldc.upenn.edu
Language Log » English is a Dialect of Germanic; or, The Traitors to Our Common Heritage
You’re right, but the Chinese word got translated to “dialect”. Most dialects are mutually unintelligible but share a common written form, which may not go both ways. A Cantonese speaker can read Mandarin using Cantonese syllables but it sounds formal; Mandarin speakers can’t read written Cantonese.