Despite the original Super Smash Bros. and supplementary material for Donkey Kong 64 pushing the notion that Diddy Kong is Donkey Kong's nephew ("little nephew wannabe"), most of the canon suggests that Diddy is simply DK's protege and not related to him by blood. 1/4 https://twitter.com/sethmacy/status/1353713810188738576
There are two branches of Kongs that are actually "families" by birth or adoption: Cranky, Wrinkly, and Donkey (most commonly referred to or thought of as gorillas), and Dixie, Tiny, Chunky, and Kiddy (species designations here get a little more complicated and controversial) 2/4
The common understanding is that "Kong" is a surname for any anthropomorphic monkey or ape (gorillas, orangutans, etc) from Donkey Kong Island or surrounding islands. However, they don't even need to be aligned politically... see Manky Kongs or the rogue Kongs of Jungle Beat! 3/4