Donkey Kong is indeed his name and he is part of a legacy. The Donkey Kong present in Donkey Kong Country was originally stated to be the grandson of the original Donkey Kong, now called Cranky Kong, and was presumably the son of Donkey Kong Jr. Donkey Kong 64 retconned this, and said that he was the son of Cranky Kong, a grown-up version of Donkey Kong Jr. And then, they switched him back to being a grandson for Super Smash Brothers and Donkey Kong Country Returns.
"One swift blow was all that was needed to leave him sprawled on the floor, face down! he groggily rolled over to see the familiar wrinkled, white-bearded, grouchy face of his old granddad 'punky Kong' peering down at him." - Donkey Kong Country instruction manual, pg. 5–6
"In his heyday, Cranky was the original Donkey Kong who battled Mario in several of his own games." - Donkey Kong Country instruction manual, pg. 6
"Well, if it isn't my lazy, good-for-nothing son." - Cranky Kong, to Donkey Kong; Donkey Kong 64 in-game dialogue
"As far as I know, 'our' DK is the son of Cranky, which does indeed make him the original DK Jr. all grown up: so if you see Cranky referred to as DK's granddad anywhere, just cover your eyes and hum loudly until it goes away.)" -Rare.com interview
Nintendo. Super Smash Bros. Brawl. Level/area: Codec conversation between Snake and Otacon. Otacon: "The Donkey Kong who fought that epic battle with Mario was this guy's grandfather."