IMO, the key to understanding Harry Potter, Neo, and James Bond -- versus Rey -- is to understand that Harry and the rest were all created alongside their respective universes. In fact, the universes were designed in large part to accommodate or house those characters. This helps them to seem "in place" in their universe, even when they are unusually skilled and/or competent.
The thing about a Mary Sue, usually, is that the character sticks out as being unusual in that universe. The term and idea originated (as far as I remember) in Star Trek fan fiction, where an author insert character would be smarter/more logical than Spock, a better engineer than Scotty, win the love of Kirk, broker peace with the Klingons, etc. The Mary Sue was universe breaking, in this way, and diminished everything around her in comparison.
It's hard for a protagonist to do this in the universe created to house them, usually, because however skilled the protagonist is, the antagonists are (almost) equally prepared, and/or the story is set up in a way to challenge the protagonist internally as they go on their hero's journey. (And if they are not, it is unlikely for a story -- or a story universe -- to become popular enough to be worth this debate in the first place; it will lack the necessary drama to inspire peoples' interest.) Harry is (or becomes) a powerful wizard, true... but Voldemort is even more powerful, and Harry must deal with the pressures of growing up, coming to terms with his family, etc. Neo is the chosen one, true... but he is up against the entire world, in a sense, and a bunch of confusing metaphysical mumbo jumbo, too, and then he must learn to master himself in order to master his surroundings, and so on. James Bond is the most competent spy and action hero ever, yes... but his villains have egos and technology and henchmen to (nearly) match his efforts, and there are political machinations, and so forth.*
Contrarily, Rey stands out because she comes in and seems so much more powerful than everything around her, in comparison, and because everything seems to come so easily to her (she does not seem to struggle to the same degree to realize her power that a Harry Potter or a Neo does, for instance, though I will note that James Bond also tends not to have to struggle very much... or at least, not in the earlier films in the franchise). We can compare her skills against the people she's next to, of course, but also those who came before, like Luke. And she just seems better than everyone else, at virtually everything she does. Everyone past or present, protagonists and antagonists alike, seem to pale in comparison to Rey's "natural ability"... which does tend to play havoc with the stakes and dramatic tension.
Perhaps this will be redeemed in the third film of the new trilogy? I find that the first two movies were different in major respects, so much so that they don't really cohere very well, or tell a unified story. It seems to me that Rey could be anything or anyone in the third movie, and give fresh arguments to either side in this (unfortunately and unnecessarily politicized) debate. But based on what we've seen so far, and especially the first movie, I would say that yes, Rey is a Mary Sue. Or she is a badly drawn, tension-deflating, universe-breaking character, at least, if the specific term "Mary Sue" offends.
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* To understand better what I mean by "comparison," imagine instead that Harry Potter had actually, originally, been a story about Ron Weasley... and everything was designed around his level of competency... and then, later on (in a sequel, perhaps), Harry was introduced as the new protagonist, and he comes in as a fully fledged wizard with Dumbledore's skill, or even more, and blows everything up, including whatever piddling problems Ron was working on at the time.
The universe, having been designed to highlight/test a person of Ron's capability, would not be ready for Harry Potter (a fully functional Potter, if you will -- not the unsure, untested kid/teen of duration of the actual series) as protagonist. He would make everything seem silly and less than, in comparison. Arguably he already does this with respect to Quidditch, which I have long felt was poorly designed on Rowling's part, but if the series had been designed for Weasley, Potter's entrance would make everything look ridiculous.
