One thing to keep in mind is that Diana was wildly popular with people who didn't otherwise pay much attention to royals. Especially when you're looking at her fan base in the US (and possibly other places outside of the UK), an awful lot of people who considered her "their princess" couldn't have named or recognized another princess until Sarah Ferguson came onto the scene. She will always have a special fascination to those people that goes far beyond that of Victoria, Mary, Letizia or Maxima for the simple fact that many of those folks simply don't know that those other royal women exist. Her wedding looked especially like a "fairytale" because it was the only royal wedding they'd ever paid attention to.
Which raises the question: why Diana? I think a lot of it comes down to timing.
Diana was introduced to the world at the end of a long recession. The news had been pretty dark and depressing for a while. In particular, what we in the US were hearing out of Britain was almost entirely about things like labor union unrest and IRA bombings. The Iran hostage crisis hadn't been over long. Earlier that year there had been an assassination attempt on our president. People were just generally hungry for something bright and happy and fluffy. Every major TV network carried the wedding in the US (and remember, cable was new and still wasn't even available everywhere, so blanket network coverage meant there weren't other viewing options) and there it was: one of the grandest, largest-scale royal weddings to date. A real life prince and princess riding in fancy carriages and wearing clothes that looked like something out of Cinderella. Princess Anne's wedding had also been televised, but that was years in the past and not at a moment when international audiences were more in a state of mind of dealing with the (not insignificant) problems of the day head on rather than staying tuned in and just enjoying something so...escapist...as the fantastical, almost unreal spectacle that Charles and Diana's wedding was.
What's more, it was the first major BRF wedding in a very long time in which the bride was the one marrying into the family. That added extra fodder for girls and women to enjoy the "what if I became a princess" daydream as they watched her, and it was a new experience to see that happen.
Then, as the economy improved and pop culture went through a phase of extremes, Diana developed a sense of style that captured non-royal-watchers' attention in yet another way. If her wedding provided a head-in-the-clouds moment of imagining yourself in her shoes, what came next was a much more attainable kind of Diana-as aspirational-figure stage that cemented interest in her.
Now, royal life and duty is not about fashion. I don't think at all that it is. I doubt her fashion was any more important to true royal watchers of the day than Charlene's, Victoria's or Kate's clothing is to us now. I don't think it was any more important to Diana than it is to any of those women today. But I do think style had a very big role to play in the development of the Diana's very intense following.
First of all, she caught attention by not being so unapproachably preppy as the royals could tend to be. She was elegant, but in a modern way. She was visibly interested in the pop culture trends, but she put a wholesome spin on them. We were seeing a lot of "bitchy" glamour on evening soaps, and she didn't give off that air at all. The younger trends were very much caught up in pushing the limits of brashness and weirdness (think punk rock, New Wave, hair bands, boy-toy era Madonna). She offered an image of how to look young and fresh without cutting and teasing your hair into an asymmetric concoction. There weren't really a ton of examples of that middle approach being pulled off well, so she continued to get a lot of attention from, again, people who otherwise would have ignored royal goings-on.
Just about every issue of American women's magazines like Redbook and Good Housekeeping had pictures of Diana in their style sections. At my small town's public library, the rack where they featured especially popular books usually held several photo books that consisted of nothing but pictures of Diana. And it was usually "everyday Diana:" at polo matches, visiting hospitals, etc., not dressed to the nines in jewels and gowns. When she wore that red sweater covered in white sheep (and one black one), knockoffs showed up very quickly in most American department stores. Throughout the 80s, he offered a look that seemed so contemporary and stylish but also very wearable by the average woman and a lot of women took careful and continual notice of it because they felt sort of left behind by the over-the-top styles seen on celebrities and in fashion magazines.
So by the end of the 80s, when the cracks in Diana's life started to show, that news was being recieved by a fan base that felt strongly attached to her and who saw themselves in her. They had never really paid any attention to any of the other players in the royal drama of her marriages dissolution, so they weren't inclined to side with them. And in some ways, initial news of the ugliness only reinforced the idea old of fairytale Diana. They'd been introduced to her as Cinderella at the ball, now they could see the other side, the equivalent to Cinderella scrubbing floors at the whim of her mean stepmother and stepsisters (with Charles and Camilla playing those roles in popular imagination).
And then timing, again. In the 90s there was a lot of interest in the idea of women speaking up and taking control of their own lives. In that context, Diana as the one who dared to fight back seemed an act of strength that was really in tune with the zeitgeist. And when she emerged from the divorce more in-your-face glamorous, it was that fairy tale again, but with a 90s spin: Cinderella reclaimed her lost glass slipper for herself.