Snow White yearns for more than Prince Charming to rescue her. She wants love, understandable given that she lost her parents. Her household chores when coming upon the dwarves' cottage are voluntary: she empathizes with what she thought were parent-less orphans.
Cinderella is trying as much as she can to buck the system in an abusive environment: one that isolates her from others and in an era that doesn't offer much opportunities for women other than housekeeper and governess. Her sarcastic rant at a bell ringing, the fact that she frees mice from traps and dresses them, and her arrival at the ball are acts of rebellion.
Aurora is a victim of social conditioning: naive and isolated by understandably over-protective guardians that haven't allowed her to develop outside of the idealistic persona they've held her up to.
Ariel is an outcast yearning to go somewhere where others will understand her and she can gain knowledge about a brave new world, she's plenty interesting, and she's willing to stand up for herself (she did kill the eels of Ursula).
Belle doesn't suffer Stockholm Syndrome and is a proud non-conformist who is brave enough to stand up to the popular town misogynistic hunk and a beastly Prince. Her actions show Gaston (at least to the audience), just how bad he is and encourages change in the Beast.
Jasmine being allowed to pick a suitor to be married to was progressive for her time and place. note Note that she is still required to be married by her birthday regardless, which is stated to be in three days' time from her introductory scene, after which she runs away, is returned by the royal guards, develops feelings for her suitor when he reveals himself to be the boy she previously met and opened herself to, and has to help defeat the Big Bad, by which time that time must be almost up. She is intelligent, courageous, witty, and kind; she doesn't easily give in to Aladdin's/Prince Ali's courting of her so easily. She is also a terrific actress.
That is just the ones whose characterization is relatively simple and has more straightforward writings. Is Mulan's motivation for disguising as a man and ran to the army is for her father or for herself? Some mix of all of the above? (this was also brought up in-universe by Mulan herself) Is Merida a normal teenager who made a mistake or a horribly spoiled brat? Is Elsa's power an allegory for mental disorder, her sexuality, something else or all of the above?
While many Disney fans feel that a lot of the hate they have received is overblown and inaccurate, others admit that they have gotten annoyed with how the princesses almost always seem to be the ones in the spotlight while other types of Disney characters—such as the princes, female characters who are not princesses (such as Alice, Esmeralda, Megara, etc.), or even female characters who are but just haven't been included into the Disney Princess franchise (such as Princess Eilonwy, Princess Kidagakash, or Princess Elena)—haven't gotten nearly the same amounts of attention.
Ariel. Fans are a bit divided on whether she's "an idiot for throwing her life away to be with someone she doesn't know" and should be regarded as poorly as the previous three Disney Princesses (who haven't aged well), or she should be commended for being "the first Disney female lead to have an actual personality and be the one to save her prince first."
In a similar pattern to Ariel, Merida is criticized for her impulsiveness and general teenage attitudes. Some find her to be refreshingly normal and realistic for her age as well as a certified badass, others find her to be just an insufferable brat with a remarkable callous streak.
Pocahontas. Fans either think she's a decent role model who stands out due to the Prejudice Aesop of her movie despite all the glaring historical inaccuracies, or an incredibly shallow character who has little personality, delivers a preachy message and is an insult to the real Pocahontas. There are also people who think that while she is a passable character, she, and her movie, is a total letdown when compared to the previous installment's incredibly flawed protagonist.
Belle. When her film debuted, she was lauded by feminists and film critics alike for her intellectual pursuits and showcasing more agency when compared to the previous princesses. Years later, her character received a re-examination, and more criticism was directed towards her. Some felt that her film "glorifies Stockholm syndrome" and that her relationship with the Beast could be read as abusive. Other classic princess lovers/princess fans in general also aren't terribly fond of Belle, since they see her as paving the way for "princess criticism" that's so common nowadays, which lead to their faves getting harped on nonstop. Others dislike the Not Like Other Girls vibe they feel her character gives off in the movie. Still, others still think she's a good character with a unique role in the lineup, find her reading hobby "relatable", and like her unconventional romance with the Beast. And there's others who still like that her film was the closest that Disney ever got to winning a Best Picture Oscar.
Over the inclusion of characters who do not seem to fit with the line as easily as characters like Aurora or Rapunzel; that is, Pocahontas, Moana and Mulan, because they are not actually princesses. Mulan is not royalty at all and Pocahontas and Moana are both daughters of a chief, and their cultures lack the connotations of princesses. On the other, even Maui points out that Moana fulfills the qualifications, brave, strong, sings, has an animal sidekick, and her title is the closest to a princess anyway. Similarly, a lot of people feel that Elsa becoming a princess is insulting somehow to being her queenship, while others feel it doesn't matter, because she was a princess before she was queen. Lastly there are people who feel like Merida shouldn't be a princess, because she is from Pixar, while others think it doesn't matter, since Pixar is owned by Disney.
Alternatively the exclusion of certain characters is this. For example, Esmeralda from The Hunchback of Notre Dame (who actually was a Princess character early in the franchise) or even Nala from The Lion King (1994) are fan-favorites but are not in for different reasons, however people often point at non-royalty like Mulan and question this.
Disney Palace Pets is mocked by much of the older fanbase. The designs and premise itself is controversial, though people do tend to enjoy the various unused concept designs.
On a more interesting point, there's the idea of whether Sofia and Elena should be included in the lineup. This stems from how the latter is a spinoff of the former, which is connected to the franchise due to the cameos the princesses made in various episodes. Both Sofia and Elena fit the mold of brave, altruistic princesses who are independent and caring, and are generally well liked. However, there's also the issue of them being exclusive only to tv, and never starting out in theaters. One side thinks that a princess has to have a theatrical movie to be considered part of the lineup, especially since Sofia is much younger. The other side thinks that their cultural heritages and diverse backgrounds should qualify them for the lineup regardless of theatrical status.
On trivial matters, there's the discourse on what colors the Disney Princesses should be associated with in merchandise — Cinderella's color is blue... except it was originally meant to be silver until Disney color-corrected it in later editions of her movie to match with Disney Princess merch, and much like her godmothers, should Aurora's color be pink or blue?
To add more to the discussion of 'Which princess should be in the line?'', Disney realized that it'd be for the larger benefit to add more nonwhite "princesses" to the line, even if their royalty status is dubious. A lot of youtube video essayists believed that Disney was widening the definition, since the concept of a princess as we know it is based on Western European conceptions, and if the company limited themselves to only follow the 'strict rules' as the fans often wanted for years, we'd only have white princesses in the lineup.
Critical Backlash: Many of the users on certain social media websites, such as Tumblr, grew up watching these movies as kids and were inspired by the princesses due to them possibly being the first iconic female characters many girls ever came in contact with. Many criticisms of the princesses will be shot down by enthusiastic fans ready to defend them and correct any inaccurate argument.
Critical Dissonance: Despite its critics, the franchise is one of Disney's biggest and most profitable.
Ensemble Dark Horse: A 2016 poll ranked the popularity of the Disney heroines. Out of the princesses, Cinderella and Aurora - the two most often attacked for being 'anti-feminist' - placed first and third respectively (Elsa from Frozen placed second).
"Cinderella didn't do anything, everything was done for her by the mice and her fairy godmother, and she only got a happy ending because of a man." is countered with pointing out Cinderella was an abuse victim who managed to escape because she had friends to help her (friends that she had made by being kind), and as a bonus, she gets to live a happily ever after with someone who loves her.
"Ariel did all of it for a boy!" is countered with Ariel already having an interest in the surface world by the time she met Eric and that he was just the icing on the cake, and she was also dealing with an overbearing father that in an act of borderline abuse, just destroyed everything that was meaningful to her.
Fanon: Honey Lemon fans believe she could be part of the lineup, even saying she's probably related to Rapunzel to confirm her royal blood.
Ho Yay Shipping: The Periphery Demographic of older fans includes a fair number that like pairing the princesses up with each other instead of with princes. Some of the most popular inter-princess pairs are:
Minority Show Ghetto: It was noted that the darker princesses' merchandise didn't seem to sell as much as the white princesses'. Whether this was due to the popularity of the movies themselves, or that less merchandise over all was produced is up for debate.
Moe: Cuteness is often heavily associated with the princesses, as many of them are The Ingenue or have qualities like that. Snow White and Rapunzel seem to be the ones who provoke this reaction the most.
Most Wonderful Sound: Most of the princesses have beautiful singing voices and have their vocals provided by top drawer talent. As a result, their songs in their movies are usually quite pleasant to listen to.
Older Than the Demographic: The characters are all teenagers and the original films are aimed at general audiences, however the Disney Princess line is firmly aimed at little girls ten and under.
Periphery Demographic: While the franchise started out targeting young girls exclusively, Disney eventually realized the untapped market of *older* fans who had grown up with the princesses and wanted merchandise for themselves who had always been there to buy some of the more reserved dolls or prints. Now Disney has whole lines catering to them, to expensive fashion dolls, women's clothing, and more.
Sweetness Aversion: The extra merchandise and sequels are loaded with saccharine morals, songs and messages that tends to be unpalatable to anyone but very young girls.
Testosterone Brigade: The especially Ms. Fanservice characters—Ariel, Jasmine, and ostensibly Pocahontas—invite a lot of male fans, but many little boys grew up with crushes on one of the princesses that stuck. Mouse Madness suggested Disney might have a market if they made the princesses just a bit more welcoming to boys, since many of the princesses are positive role models and many have traits boys can relate to. A good gateway would be Mulan, since she acts like a boy for half the movie.
Many of the girls coloring and style is changed for the merchandise, such as Aurora's lighter hair, Cinderella's dress changing from silver to blue, her hair changing from strawberry-blonde to bright blonde and so on.
After Disney unveiled new hairstyles and dresses, some people weren't very happy about that.
Aurora and Tiana graced the wrappers of some dipping candy. Aurora got the packet of vanilla-flavored sticks. Tiana got the packet of watermelon-flavored powder. This was met with some amount of controversy.
The new designs attempted to homogenize the general look of the Disney Princess line and the different art-styles, which led to everyone either getting paler or darker, the brightness being turned up to max, and some of the efforts to make the art seem more detailed really failed to the point where dark-eyed characters seemed blue-eyed. The homogenization of art style, however, led to the faces looking more Caucasian than anything. This sparked a pretty big backlash even outside of regular fandom circles.
At least in Japan, most of the merch only focus on the first six (and Rapunzel), with the others getting little to none, which might reek of bias due to the others being princesses of color (Save for Merida, but why she gets that too is unknown).
One justification of the Real Women Don't Wear Dresses criticism. Not only were the original fairy tales written centuries ago, but the time periods in which the earlier Disney Princesses were made had the "demure-but-hard-working ingenue" type as the epitome of womanhood in mainstream America. Naturally, this flies up the head of many modern fans who just peg them as "weak" and "whiny".
Ever wonder why Japan only recognizes 7 out of 12 princesses in the franchise there unlike the rest of the world? Apparently, one Quora user found out the reason stated in this link that the Japanese have their own list of qualifications to have one recognized as part of their princess canon, as well as having an old school vision of what a princess is for them.
All of the Disney Princesses are treated as such. The older princesses receive the most disdain, as the time period their movies were made from put them in the passive role that would not be acceptable in a female role-model today. Even the modern princesses are often closely scrutinized and found unworthy, particularly Ariel. While there is some truth to the criticisms, the biggest criticisms lobbied at the girls tend to twist the actual events of the movie. For instance, the common criticism that Ariel gives up her life at home for a man isn't exactly true—Ariel clearly desired to live with humans long before she knew who Eric was (her famous musical number was even all about it!), he just was the catalyst to actively going out, as well as blaming Snow White and Aurora for "not saving themselves", despite the fact that both were under spells against their will and could do nothing about it.
Disney draws a lot of ire from the older fans when Mulan is put in a feminine dress (particular the pink one with make-up that she felt uncomfortable in). However, as with the above case, even those criticisms are often twisted—Mulan is also clearly uncomfortable wearing armor and hiding herself that way, and did like looking cute in the pink dress even if she disliked the prospect of an Arranged Marriage. The happy medium is usually the far more practical (and plain) dress she wears at the end of the film.