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?
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.
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Same Person different name. This name is Ivo Robotnik, he's called "Eggman" because of his shape. Really a nickname that's been with him since Sonic adventures.
I don't remember him having a preference or not liking the name "Eggman". He like some of the characters, does have alternate universe version of themselves. Like "Dr. Eggman Nega".
At the risk of showing My Age and Geekiness... (Also My Awesome Memory.).
Just before, or as Sonic The Hedgehog was being released on the Megadrive, a Publication called, "Mean Machines Sega", (UK) had a pull out Comic. http://www.recalledcomics.com/SonicTheHedgehog1UKPromo.jpg
(Sonic The Comic didn't appear, for a couple of Years after that.).
In this pull out Comic, Sonic starts out as being Brown, (The Cobalt turs Him Blue... Why there's Cobolt involved in a Treadmill, I have no Idea,) in the explosion Doctor Kintobor turns evil for some Reason. -To be fair, it was a really flakey Story.
So, being His Polar opposite in a Flakey Story, (Much like the Kaleds became Daleks, for all of You WHOligans, out there,) KintoboR became RobotniK.
After typing, [Kintobor] into a search Engine, I have found this Image. I hope that it does not breech any Rules of posting.
So, based on that: His name is Robotnik. Although, I suspect that His original Name was Eggman.
Ivo Robotnik and Eggman are actually one and the same. Check this link for references.