Aladdin is a strange movie to discuss in the context of Disney Princesses. Unless I've forgotten something, it's the only official Disney Princess movie where the princess isn't the protagonist. In fact, although she has a couple of scenes without Aladdin, Jasmine's role in the movie isn't that significant, and she's absent for many of the important scenes. But I think it's interesting to look at how Disney treats its female characters when they're not the protagonist of the story.
And it looks like Disney put a lot of effort into giving Jasmine "girl power" and independence, at least in her dialogue and attitude. Unfortunately, they didn't follow through and give her strength in the plot itself.
Jasmine gives a lot of lipservice to being Strong and Independent. She's determined to marry for love, and is slowly but surely driving off all potential arrogant suitors. She yells at the male characters for trying to decide her future, declaring that she is not a prize to be won, she calls Aladdin out on his lies, and she runs away from the palace so that she can make choices about her own life. She's a fun character to watch, for sure, and I think she's intended to be a strong female character who contradicts the sexist assumptions and stereotypes of some of the men around her -- Jafar's statement that he likes his women "speechless," for example, and the way so many male characters try to "win" her. She even tricks Aladdin into admitting that she's a "fine prize for any prince," and shouts at and rejects him when he agrees. And this is all great stuff.
But there's not much to her beyond this lipservice. She wants to marry for love, but we don't really know what else she wants. She's spent her life in the palace, but we don't see evidence of any hobbies or interests, unlike her father. She doesn't have any friends or acquaintances or even servants beyond her father and her pet tiger. And sure, Aladdin doesn't really have anyone except Abu, but other people in the city seem to know him, and we get a much stronger sense of his personality and struggles. More importantly, he actually gets to control his own life and experience character development. Jasmine has no such luxury.
When we first meet her, she's being told that she must marry in the next three days, not only because the law demands it, but because her father wants to know she'll be looked after once she's gone. And at the end of the movie, she calms his fears by finding someone he can trust to take care of her, and he changes the law so that she can "choose" him. He won't change the law so that she doesn't feel she needs to marry, of course, or so she can rule. But he'll let Aladdin be the new sultan if that's what it takes for Jasmine to find a husband. And Jasmine is lied to and manipulated every step of the way. Jafar tells her that Aladdin has been executed, so she not only thinks Aladdin is dead, but thinks his death is partly her fault. Not the most healthy basis for a relationship when she eventually finds out that he isn't dead after all. And of course, she doesn't call out Jafar for his blatant lie once she realizes that Aladdin and Ali are the same person. She does call out Aladdin for lying to her, but he immediately weaves more lies instead of telling the truth, and in the end she forgives him without any further explanation.
Jasmine is also the only named female character in the entire movie. A couple of other female characters speak or sing, but they're either negative caricatures of unattractive women, or they're belly dancers. Every single one of them. In the end, even Jasmine is put into this role, and has to use her sexuality to attempt to save the day. I guess her intelligence and ability to read people that's been clear for the rest of the movie wouldn't be useful here. And then she fails and is trapped in an hourglass, forced to wait for Aladdin to sweep in and save her life. Just as he saves her from losing her hand in the marketplace, and saves her from being forced to marry Jafar. Just as he's the one to finally "show her the world," in one of the best sequences in Disney history.
Of course, Aladdin is the protagonist and Jasmine is the love interest, so it makes sense that he gets to perform the significant heroics here. Except this is inconsistent with previous Disney movies. In Snow White and Sleeping Beauty and The Little Mermaid and Beauty and the Beast, the male love interests fight and destroy the villain, while the protagonist plays the damsel in distress, or, at best, runs to try and help without actually doing anything to help at all. But in Aladdin, the protagonist fights and destroys the villain, while the love interest plays the damsel in distress. Strange how that works.
Aladdin is another gorgeous movie with fantastic music and lots to enjoy. The theme of "be yourself" is a good one. And it's definitely progressive in lots of ways -- including Jasmine's response to her suitors and her situation. But it isn't quite as progressive as it thinks it is.