Not hackers. Just trained.
May be considered the most intelligent overall, and/or most creative, but that's it.
Anyone can use an omni-tool. Engineers however focus on quick, efficient, useful, amazing, impressive (etc etc) ways with it. Its a certain sort that can actually achieve miracles through technology, instead of just following its lead or hitting it and yelling "Stupid machine!" Engineers don't just work tech, they use tech to 'hack the world' around them. As seen through their abilities: Incinerate, Overload, Cryo Blast, Combat Drone, Sabotage, Sentry Turret, AI Hacking. And for MP: Geth Turret, Hunter Mode, Homing Grenade, Arc Grenade, Supply Pylon, Tactical Scan, Arc Grenade, Energy Drain, Decoy, Cain Trip Mine, Concussive Arrows, Armor-Piercing Arrows, Proximity Mine, Recon Mine, Submission Net.
Would count abilities of squadmates with Tech too (Tali, Garrus, Kaidan, Miranda, Mordin, Legion, Kasumi, EDI, maybe more), but I'm tired and I think they're at least mostly covered by the ones already listed.
Anyway, the point is that Engineers just take the technology that they or others develop, and put it to best use. A Soldier could Hack AIs, sure, but they'd probably suck at it. An Adept could set up Mines, but they'd probably suck at it. Overall. More likely. Beyond that, this is just a class archetype and individual characters can be whatever they want. Indeed, that's kinda what happens with Shepard and his Bonus Powers.
Engineers came to be though the same basic routes as Adepts or Soldiers: They were created with it or they tried really hard to get to it. Engineers just focus more on technology, not strength or skill. They may have high levels of these two things, of course, but as a class, they prefer to make the most use of the technology that suits them. Technically a 'Soldier' with tons of ammo powers and an in depth knowledge of them and how they work, would be at least akin to a sort of 'Engineer'. But most Soldiers with ammo powers use grab and use them and more on with shooting with as much will to fight as they can muster. For example, I wouldn't expect an Engineer to be too useful compared to a Soldier, if the Engineer's technology was stripped away from them. But let them have the tech, and they are masters of that domain.
Shepard as an Engineer is just a Shepard that could be considered to be relatively bright in that field. IMO his roleplaying in at least most of the story and its setup is that of a Soldier, but so is most of the Alliance. We get exposure to more 'extreme' 'Engineers' though characters like Legion and Tali and they'd apparently go beyond what most Alliance personnel would be (particularly when it comes to AI Hacking), and Shepard specifically revolves around the world 'soldier' and did training that oriented him around 'command' and high risk 'combat' scenarios. Broadly, I'd consider Engineers (though ME1-3 doesn't care about this so much) to be the sort to actually try to avoid combat whenever it seems possible. Not necessarily because of cowardice, but just because its considered that technology can solve problems, including solving conflicts before they may even start.
So that's what I think of Engineers. They engineer. They like the concept of applying knowledge, intelligence, and technology to solve problems. The origin of an Engineer probably revolves around that concept on some scale (personal, cultural, etc).
The Alliance uses tech and faces a galaxy that needs quick and powerful use of tech in combat. So they train engineers that don't just use their skills when there is peace. That's the kind of thing that the omnitool brought to the galaxy.