"A language is a dialect with an army and navy"
Unfortunately, there is no hard scientific distinction between "two related languages", and "two dialects of the same language".
A biological analogy might be when are two groups of animals considered distinct species? Inability to interbreed (cf. "low mutual intelligibility") is one criterion, but it is not necessary for two groups to be considered distinct species e.g. chimpanzees and bonobos.
Likewise, while it might be easy to categorise two lects with low mutual intelligibility as distinct languages/belonging to distinct language groups (e.g. English/Japanese, or French/Italian), two closely related lects may be either defined as distinct languages or dialects of one language, depending on historical and political context (e.g. Portuguese/Galician, vs Catalan/Valencian).
There is no consistent, scientific distinction between two dialects of a single language, and two separate languages (with a common ancestor). It's most often a political distinction. Some highly mutually intelligible lects are considered different languages (e.g. Serbian and Croatian) whereas others with extremely low degrees of mutual intelligibility are considered the same language (e.g. the Valais dialect of Walser German and Standard German).
- Is Occitan a single language or have its different dialects become separate languages? (linguistics.stackexchange.com)
Chinese dialect groups
Local varieties from different areas of China are often mutually unintelligible, differing at least as much as different Romance languages and perhaps even as much as Indo-European languages as a whole. These varieties form the Sinitic branch of the Sino-Tibetan language family (with Bai sometimes being included in this grouping). Because speakers share a standard written form, and have a common cultural heritage with long periods of political unity, the varieties are popularly perceived among native speakers as variants of a single Chinese language,[58] and this is also the official position. Conventional English-language usage in Chinese linguistics is to use dialect for the speech of a particular place (regardless of status) while regional groupings like Mandarin and Wu are called dialect groups. ISO 639-3 follows the Ethnologue in assigning language codes to eight of the top-level groups listed above (all but Min and Pinghua) and five subgroups of Min. Other linguists choose to refer to the major groupings as languages. Sinologist David Moser stated that the Chinese authorities refer to them as "dialects" as a way to reinforce China as being a single nation.