Eggman was the name SEGA came up with, but back in the earliest days SEGA worked closely with their American branch and formed a team called "STI" to develop Sonic. As the goal was to create a mascot that would appeal to American audiences, as that was the most profitable market and that demanded a new competitor as Nintendo was the only game in town in a literal sense.

The Japanese branch wanted a very specific story about an American woman who created Sonic The Hedgehog as a children's book character based on her late husband (a fighter pilot), only for Sonic to be real one day, apparently her husband reborn on "Christmas Island"... With Sonic having Eggman as his big rival, with the two more or less being Bugs Bunny and Elmer Fudd in their characterization. There was also the idea of Sonic running a band with characters called Sharps and Vector (who was later recycled for the game Knuckles Chaotix), but that was veto'd early on when the band wouldn't fit on the Sonic 1 cartridge.

America basically stood up and said "That story is weird and very little of that actually translates into a game format" and came up with a different plot where Sonic was from the planet Mobius and fought against the evil Dr. Robotnik who had conquered the planet, with Sonic having Freedom Fighter pals who helped him on the fight.

The Japanese portion of STI basically rebelled against the American portion for actually doing the job they were hired for (Americanizing their Sonic idea, which was designed to be Americanized to begin with), kicked them off of Sonic 2, and delegated the Americans to spinoffs like Sonic R, Sonic Spinball, Dr. Robotnik's Mean Bean Machine, and a game called Sonic Xtreme (that was ultimately cancelled because the Japanese

SEGA is a highly xenophobic company, and their history of spiting the American Branch extends far past the Sonic series (With one of the more infamous examples being SEGA demanding Eternal Champions' sequels, toy line, and cartoon be cancelled because the Virtua Fighter director was salty that Americans liked Eternal Champions better.... even though the reason was because Eternal Champions was a home console game at a time when Arcade Games like Virtua Fighter were losing favor...)

Robotnik was NEVER a robot (Outside of the early Archie Comics where they mistakenly labeled him as one a couple of times, but that's hardly the oddest thing in Early Archie) in any continuity, the reason the name Eggman was changed to Robotnik was due to the American Team warning the Japanese Team that the name "Eggman" would not be taken seriously as a villain name and would have to be changed. Dr. Robotnik was what they picked since "His thing is robots, not eggs."

It never had anything to do with censorship

The American Team was right as when Robotnik's name was changed to Eggman in the US, many outlets complained that the name was absolutely ridiculous and the fanbase largely refused to use that name until younger people who grew up with the Eggman name entered the fandom. Now even in fan circles the name Eggman is more common except for the one or two grey muzzles like myself who insist on Robotnik and that the dang kids need to stay off our lawns. (Even a recent and popular fan-game Dr. Robotnik's Ring Racers which has the name Robotnik in the title still calls him Eggman in the actual game)

The American Plot has largely vanished with occasional elements of it cropping up from time to time (With the Freedom Fighters now called Team Sonic and excluding all American Made additions like Sally Acorn and Bunnie Rabbot, Mobius now just called "Sonic's World", Sonic liking Chili Dogs, and the name Dr. Robotnik occasionally showing up in promotional material and the Live Action Films)

tl;dr - The name Eggman was rejected by the American Staff hired to do the Marketing because it wasn't a good villain name, then Sega of Japan insisted on putting the Eggman name back when the story was being told directly in actual games instead of being poorly parroted spinoff comic books and cartoons. Robotnik/Eggman himself was NEVER a robot, with the exception of Robo Robotnik/Eggman from Archie Sonic, but that's another continuity altogether.

(In the Archie Sonic continuity, they were adamant about never doing reboots until they were forced to, so... in order to justify the name and design change from the American Robotnik to the Japanese Eggman, they just wrote that Robotnik and Eggman were two different characters and that Eggman took over when Robotnik died. Spoilers: Around issue 50 or so, Robotnik is killed off for the rest of the book)