I'm not sure what the requirements are to get into MIT, since I never even tried to get in there. Being such a prestigious institution, the Engineering world's equivalent of Harvard or Yale, it has become rather exclusive; getting in is more a matter of who you know (faculty, alumni, financial sponsors) than what you know.
You are asking about cars and robots and which Engineering field you should focus in.
For cars, you should look toward Automotive Engineering.
For robots, you should look at Mechanical Engineering, Electronic Engineering -AND- Mechatronics.
Your Aunt is mistaken about Mechanical Engineering, it's not about the engine of a car (Automotive Engineering covers that, as well as all the other parts that are involved in making the car go; powertrain, suspension, aerodynamics, braking, etc), mechanical Engineering is about, for lack of a better term, 'equipment that moves.' In Mechanical Engineering it's all about levers, joints, gears, pulleys, basically any way of transferring energy from point A to point B through moving parts, as well as knowing how to support the weight of loads being moved. (I'm an Electronic Engineer, so I don't have the correct 'lingo' to fully describe Mechanical Engineering, not sure I fully understand it myself.) I'm not saying that a Mechanical Engineering background won't help with Automotive Engineering or Robotics, far from it, Mechanical Engineering is the foundation on which almost all other Engineering is based on, since anything built in the 'real world' needs to be supported somehow or support itself somehow, and that support is all Mechanical (or possibly Structural) Engineering.