Why isn't there a geographically-based name for the Chinese/Mandarin language, as is the case with Cantonese, Fukienese, etc.?

Answer 1

A mandarin is a government official (官 gūan); the language Mandarin is literally the language of government officials. China is known for its variety of local languages (often called dialects), as you have said. In premodern times, government officials came from all over the country; to cut down on corruption, officials would be assigned posts outside their home territory. The common language for the vast country was Mandarin, the language of government officials. Non-local by design!

Answer 2

Cantonese, Fukienese are just some of the local dialogues in the greater China region. There are over 50 local dialogues because Chinese is a combination of 56 peoples. Mandarin Chinese is the only official language that all Chinese descendants speak around the world so we can understand each other.