Why does Robotnik call himself Eggman in the newer Sonic games I can understand the other characters calling him that but Robotnik himself

Doctor Eggman was his original name, in the Japanese version of Sonic the Hedgehog. Robotnik was his name for english-speaking countries.

Yuji Naka has explained that 'Robotnik' is his real name, and Doctor Eggman is an alias that he goes by, so it's not an insulting nickname other characters have for him, it's something he himself uses, and he uses 'egg' for the names of some of his machines, such as the Death Egg and Egg Emperor.