I'm back, I pulled an Abe Simpson. Except I came back in.
Anakin it's explained that well, his owner had him do mechanic things and he wanted to podrace.
Luke wanted to fly shit. Like Rey, he had his little...car thing he drove around. He also worked on his robots so he had some mechanical training.
Rey is a junker/mechanic who can recognize components, her foster parent owns ships and she shows knowledge of the Falcon.
None of these characters are inserts with unexplained powers. None of them are Mary Sues. I suppose the Force is an aspect of Mary Sue-That is, it grants some power to its users. But either way, the characters themselves are established as having skills.
Here's the difference -
Anakin knew how to podrace because he actively built podracers and was a mechanical genius. His ability to fly a ship was presented as a happy little accident due to him being locked in the cockpit and the ship switching to autopilot. His role in the battle of Naboo was pretty dumb and heavily criticized, as it should be. Anakin being a mechanical genius doesn't give any explanation for him being able to fly a starfighter like an expert, and the image of this little kid being the one to turn the battle around all by himself was rightfully mocked back when TPM released and is still mocked today.
Luke worked as a farm hand but did not live in extreme poverty. He wasn't a slave, he was the nephew of the farm owner who appeared to be at least modestly successful. It's pointed out multiple times that he knew how to fly a ship, and we get to see that he's competent at best. When the movie reaches its climax, it's only then that is he able to really tap into the force and make the clutch shot. His entire story arch in ANH builds to that moment. Luke is whiny, he's unlikeable, and no one but Ben seems to enjoy his company much at all for most of the film. Luke's one big triumph in ANH is blowing up the Death Star.
Rey grows up a scavenger alone on a desert planet in the care of someone who clearly has no affection for her. She is at the very bottom of the rung in her little society and is portrayed as worth so little, her foster parent doesn't mind having her ruffed up to steal from her. Unlike Anakin or Luke, Rey grows up with no one to raise her or look out for her. She also is dead set on staying on the planet until her family return. The most we see out of her before her run in with Finn is that she lives in a plane and can take apart ships and no one cares about her. After meeting Finn, we're told she's able to pilot a ship. But it's not just that, she's not just competent like Luke. She's spectacular like Anakin. Her ability to pilot is a surprise to Finn, the POV character, and in turn meant to be a surprise to the audience. This is where I feel the storytelling falters, because unlike with Luke (and to a lesser degree, Anakin), we don't get any build up to her ability to pilot so well. It's not at the climax of the movie like with the other two characters. And unlike Luke, she pulls off some really impressive maneuvers that you'd expect from someone who has been flying for years. OK, I guess she's a good pilot. I can buy this development if I have to. But as I said, this isn't the climax of the film. The surprise abilities keep coming for her. She outsmarts Han Solo in his own ship and fixes the Falcon. She saves Finn from a tentacle monster that had previously been killing everything it touches. She overpower's Ren's mind probe. She masters force persuasion, something she's only heard about in stories, in about a minute. She evades detection in the heart of Starkiller base (except when she's magically spotted by Finn when the plot requires). She force pulls a saber from the snow on her first try out from the clutches of a seasoned force user. She holds her own on her first encounter using a light saber against a trained saber user. Then, the moment her disadvantage is pressed, she is able to tap into the force and overpower Ren. A moment that should have been the climax of the movie is just another moment in a long list of incredible feats. What's the moment of climax in ANH? It's not when the Death Star explodes. It's when Luke lands the torpedos into the vent and you see him breathe a sigh of relief. I personally never felt that kind of tension at any point with Rey, though I imagine many other people in this thread disagree.
And none of this accounts for some of the other ill conceived story choices such as Poe being temporarily removed solely for the sake of the plot, the Falcon being unlocked and easily stolen from a camp full of miscreants, a janitor knowing how to disable Starkiller Base, running into Phasma on a huge base exactly when they needed to without being detected by anyone else, Finn accidentally finding Rey when everyone else in the base is frantically searching for her, the bowcaster only maiming Ren when it's been previously shown to easily kill others, etc. etc. A lot of my personal problems with Rey stem from problems I had with the movie as a whole.
But all that said, I did like it! I truly did! I just didn't love it, and there are lots of reasons why. I don't think Rey is a Mary Sue. I've never made that claim. I do think her development was truncated though.