Eggman is the only human character in the original game (and would remain so for quite a few entries), so giving him a full name makes sense. As for the choice of name, Robotnik alludes to the ...

I am the Eggman In 1990, Sega was on the ropes after the messy launch of the Mega Drive. The console would be rebranded as the Sega Genesis in the US, but it needed more than that to take off.