Are they the same, or alter egos? Does "Eggman" refer to a specific persona with unique characteristics compared to "Dr. Robotnik"? Does the name change based on between the 3D/2D game perspective, or the game's region (Japan vs. US)? Does he ever assert a name preference, and does he accept or reject the Eggman nickname? They look the same—is there a reason why the Sonic series uses both names simultaneously?
The difference between Dr. Robotnik and Eggman
sonic-the-hedgehog-series
Related Solutions
Super Sonic
Super Transformation (Sūpā-ka) is a frequently recurring gameplay feature and story element within the Sonic the Hedgehog series. It was first referred to by name in Sonic Heroes.
It was known as 'Super' back in Sonic the Hedgehog 2, when you collected all the Emeralds the text would say "Now Sonic can be Super Sonic" (and again in Sonic 3 and Knuckles, Now <character> can be Super <character>"
Hyper Sonic
Hyper Sonic (Haipā Sonikku) is a hyper transformation of Sonic the Hedgehog that is obtained by collecting all seven Super Emeralds, Chaos Emeralds that have been enhanced by the Master Emerald. To trigger the transformation, Sonic must collect at least fifty Rings and perform the Insta-shield. To this date, Hyper Sonic is recognized as the most powerful form of Sonic ever.
Darkspine Sonic
Darkspine Sonic (Dākusupain Sonikku) is one of Sonic the Hedgehog's transformations, appearing in Sonic and the Secret Rings. Rather than using the Chaos Emeralds, he uses the World Rings of rage, hatred and sadness to transform. In this form, Sonic's personality is slightly more sinister and violent due to the intense feelings that each of the World Rings contain. He is playable in the final battle against Alf Layla wa-Layla, an enhanced form of Erazor Djinn.
Excalibur Sonic
Excalibur Sonic (Ekusukaribā Sonikku), also written as Excalibur-Sonic, is one of Sonic the Hedgehog's transformations, appearing in Sonic and the Black Knight. Sonic assumed this form by the power of the transformed Caliburn and the three sacred swords. He is playable in the final battle against the Dark Queen, an enhanced form of Merlina the Wizard.
Dark Sonic
Dark Sonic is a next form of sonic that was only seen in the Sonic X series. The form was gained from sonic seeing his friend in pain. The power enhances Sonic's strength, speed, and some what his anger.
These are all the transformations for Sonic as documented on the "Sonic Wikia".
While there are real-world machines which accelerate objects horizontally and release them, I think that the specific appearance is mostly meant to be legible in the context of the game: the L shape is a platform to stand on that you won't slide off (possible visual reference: ski bindings), and the round wheelish bit suggests that it is movable/active (possibly: roller skates).
Given that both of the appearances you name are on large (air)ships, perhaps it was inspired by an aircraft catapult which is used to launch aircraft from the short runways of aircraft carriers (by accelerating them quicker than they could achieve with their own engines).
As evidence that the design is at least somewhat particular to that level's theme, consider that other levels use different designs with the same function:
- Sonic 2, Chemical Plant Zone: there is a similarly-functioning item which is instead depicted as a set of spinning wheels (as might be in a baseball pitching machine).
- Sonic CD, Stardust Speedway: the machine is styled like a gun, recoiling and ejecting a cartridge when activated.
- Many other levels simply have tunnels that accelerate you without giving much explanation other than possibly arrow-shaped decorations.
I haven't played Kirby games very much so I can't particularly give citations if there are any.
Another type of real world machine which can be L-shaped, accelerates along a track, and is somewhat aeronautically themed is the rocket sled.
(image from Wikimedia Commons, public domain)
Best Answer
They are different names for the same character. The Wikipedia article you linked has the explanation, sourced from the former head of Sonic Team, Yuji Naka:
In various iterations of the franchise, they've chosen different names for him, with Robotnik being more of the English name, and Eggman the Japanese. However, since Sonic Adventure (1999) he tends to go by Doctor Eggman in both English and Japanese translations of his video game appearances.
As far as whether or not this is a preference, a teasing nickname, or just a localization hiccup, the character rarely remarks about his different names or his own preferences. However, as Jupotter points out, in Sonic Generations, Classic Tails refers to him as Dr. Robotnik, and he comments that nobody calls him that anymore. This would seem to indicate that he's at least aware of his different names.