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From the AffablyEvil.Video Games subpage:

• Dr. Eggman from Sonic The Hedgehog has his moments of this.
  • In Sonic Adventure 2 he acted very pleasant and, though he came close to killing Sonic, he was shown almost regretting it, showing profound respect for him after all the years that they were rivals. He had also been seen getting along with Tails at the end of the game. It also showed that he had the deepest respect for his grandfather, Gerald Robotnik.
  • The best example was probably Sonic the Hedgehog (2006), where he's shown acting politely and gently most of the time, especially in front of Elise (except for the whole kidnapping part, of course).
  • In the ending of Sonic Generations: The classic version, having learned that he never defeats Sonic, ponders upon pursuing a career as a teacher instead.

From the Sonic the Hedgehog - Dr. Ivo "Eggman" Robotnik page (which I actually admit to contributing to in the past, via minor rewording edits):

• Affably Evil: Despite really wanting to Take Over the World — and occasionally going to some extreme lengths to achieve this goal — Eggman's a fairly pleasant fellow who usually treats his enemies with Villain Respect.
  • In Sonic Adventure 2, he lowers his head in melancholy after seemingly killing Sonic, and later gets along with the heroes after teaming up with them.
  • In Sonic Colors, he takes Sonic's advice regarding threatening him directly rather than going through a bunch of terrible amusement-park-related puns on the way — for which Sonic then thanks him.
  • In Sonic Lost World, he saves Tails' life from the rogue Cubot, even though it isn't necessary for his master plan, and seems to genuinely respect Tails. He's a complicated guy.

Although Eggman's polite demeanor isn't constantly intimidating per se, it's still merely a sarcastic extension of his "gentleman genius" persona. He also frequently demeans Sonic the "dear spiked rat" and his friends at every other opportunity (and any "admiration" he has for Sonic is more regarding the hedgehog's tenacity), and it's clear from other tropes applicable to him that he's only out for himself. Most truly Affably Evil characters (of which the Sonic X and Sonic Boom Eggmen do count, even if unconsciously) tend to be far more open and dog-petting in their niceness, in contrast to how Eggman is selfish, misanthropic, and generally treats his underlings horribly.

He loves being what he thinks is the most civilized and enlightened guy in the room, plain and simple, and the trope subpages of quite a few different adaptations have already described their Robotniks/Eggmen as "Faux", despite how two of them (AoStH and the movie) were heavily comedy-focused.

Now, I'm sure there might be a couple cases out there of game-universe!Eggman being genuinely affable (maybe just during Olympic season or something, the Twitter Takeovers are more of a different kind of portrayal altogether), but these are too few and far between to be a defining trait.

• Sonic Adventure 2:
  • The Misaimed Fandom entry I put on the game's YMMV page explains the "Gerald" bit in detail. Long story short, he looked up to his grandpa as a brilliant scientist rather than because of anything benevolent, and he's disappointed that Gerald went crazy trying to destroy the world since he takes pride in his own Pragmatic Villainy.
  • The "admirable adversary" scene has him send Sonic to his death via explosive escape pod and then lowering his head out of respect… before immediately getting over it and pointing his gun back at Tails. His Dark Story recap also has him gleefully proclaim, "I finally did it! I've defeated Sonic! That annoying hedgehog is gone forever! He's nothing but floating chunks in space now!"
  • He did get along with the heroes during the Last Story, but, y'know, no world means no empire so necessity and all that.
  • The somber "you're right" response he gives to Tails' "we all did it together" was meant to fit with the equally somber tone of Shadow's death and Gerald's tainted legacy. As sincere as it sounds, it's just Eggman acknowledging the facts after being briefly lost in a moment of thought.
  • Here's a Dummied Out exchange that was meant to happen after that, anyways:
    Tails: "Where are you going?"

Eggman: "Next time, I will pull this off without anyone's help! I will use my own power to take over the world! Next time we meet, all bets are off! Tails: "Same here!"

• Sonic the Hedgehog (2006): He's only gentlemanly to Elise because she has something he wants and he still tries to get rid of Sonic and friends in multiple ways. And he doesn't even give Elise a nice room or a meal.
• Sonic Unleashed: He keeps Professor Pickle fed with cucumber sandwiches on request. Yeah, a corpse is less useful than someone to interrogate, food makes a good bribe in case you don't have a lot of other time and options, and he got the recipe wrong anyways.
• Sonic Colors: The "Eggman changes his evil speech at Sonic's request" scene was most likely meant to be played for laughs given how the doctor does it begrudgingly. There's also this line he gives afterwards to Sonic: "I would say it's been nice knowing you, but it hasn't." Not even this game has a genuinely nice Eggman.
• Sonic Generations: The whole "teaching degree" joke was probably meant to be sarcasm from Classic Eggman, with Modern Eggman missing the point and only considering it because "I've always enjoyed telling people what to do!", causing his Classic self to Face Palm. Also Lost World and Mania happened because each Eggman went back to world conquest, so no.
• Sonic Lost World:
  • It's stated in a Magnificent Bastard entry and supported by the game's ending that he only saved Tails to make sure his Enemy Mine with Sonic stayed on schedule. At best, I guess you can argue that he'd want to beat Sonic and Tails personally.
  • A scene earlier into the game (the one where Sonic sets the Zeti loose) has him tell the Deadly Six that he's a "compassionate" man… with the rest of what he says being covered up by Sonic and Tails talking over him. Listening closely, this is what he says to the Zeti:
    Eggman: "I'm a compassionate man, or else I would've dropped you into a bottomless pit by now!"
  • The game in general seems to subtly emphasize how is Eggman a Jerk with a Heart of Jerk— he's back to stuffing animals in robots, he only sees the Zeti as useful lackeys for him, he doesn't care that the Cacophonic Conch's noises are "very painful" to the Zeti and only that the shell itself is "very rare", and he only saves Sonic's life so the hedgehog can get rid of the Zeti for him while he sneaks back to his Extractor device. And to top it all off, this line just oozes with Faux Affably Evil:
    Eggman: "Finally, with the energy drained from the Extractor, I can rule the world! Pity about the damage done down there, but there's still enough left for me to conquer."

That pretty much covers everything here. I mean, sure, he's a very charismatic fellow, but don't let him fool you: it's all just an act. He indeed wants to kill Sonic on his terms, he only respects his enemies because they actually give him a challenge, he only acts cordial to his victims to get something out of them, and he only praises his underlings for, to quote the original Sonic Adventure, being "of use" to him and "spare parts" if they screw the pooch.

And considering how he's a very Opportunistic Bastard willing to pervert anything to get what he wants (e.g. Little Planet, Angel Island, an ancient pyramid, a planet, five planets, time and space, a guy's very reason to live, etc.), I think it's safe to say he'd even exploit a Draco in Leather Pants situation like this one if he existed in the real world.

P.S. Faux Affably Evil is currently being used to differentiate Eggman Nega from his main self. I'd say they're both that trope but in different ways— per Sonic Rush, Eggman's demeanor is more whimsical and Nega's is colder and more callous. Think something more like Megatron and Tarantulas.