As someone who has actually seen the play I would actually disagree with this pretty strongly and I'm already a little tired of this take. The play does emphasise her gender nonconformity and her insecurity, traits that are extremely Arya-like and not very Sansa-like. Lyanna is not an Arya clone, she is her own unique character, but there are key similarities there in terms of how they grapple with their own gender nonconformity.
I also think the play highlights how much this isn't a House v House conflict, unlike Starks v Lannisters, but rather a rebellion against Aerys, which if Rhaegar hadn't fucked it, he would probably be leading himself. Lyanna in the play even says that 'Robert's cause is just' in Act 2, the two refuse to engage in the fighting, and Lyanna only pushes Rhaegar to fight to protect her unborn child after nudging from Elia. So I don't think the comparison here to the War of the Five Kings fully tracks.
Here's something I wrote elsewhere:
But what I'd like to emphasise at this point is really how much I think Lyanna is an opposite to Sansa? Where Sansa is secure in her femininity and rewarded for her ability to sing and sew, Lyanna is insecure, unsure, uncertain, feels she not only wants to do these duties but honestly on some level can't (and is much more Arya-like in this sense). Sansa adheres to rules rigidly, thinking if she's just 'good', everything will be fine - whereas Lyanna can't do that, it's not in her nature. Where Sansa looks forward to marriage, Lyanna is much more uncertain, resigned to it a little, but feels unfit for it. There's nothing in the play setting up Lyanna as a starry eyed romantic looking for love before she falls for Rhaegar - she's not looking for songs, or princes, or one true love, it just hits her all quite unexpectedly. Her plot involves being a lady trapped in a tower, but in the play this is not quite about a false fantasy being smashed by the ugly reality - she never grows to hate and resent Rhaegar as Sansa does Joffrey, she is not quite a hostage, the horror that befalls her is more the madness of falling desperately in love and the worst of grief and the darkness of obsession.
She might be a character in a romantic story Sansa might find intriguing, but she's unlike her in so many ways. I mention all this because you brought it up and others have done so to me several times, but personally I find the reading a bit shallow? Like the only similarity is that Lyanna falls deeply, madly in love with someone... and really, almost anyone who is not a aromantic can do that, in the right circumstances? Beyond being in the subgenre of romance, they're operating in quite different ways - they're honestly more foils than parallels, and I think if they knew each other Sansa would regard her aunt as a bit 'weird'. Her interest in Lyanna's love story would be mediated by the social judgement that she has been taught to espouse towards women who are so unladylike, who reject society.
I also think this post is a misreading of Lyanna's loss of magic? Lyanna loses magic as she approaches adulthood and the normalcy of adulthood - having to conform to gender expectations - looms. She's resigned to this. But her magic returns with Rhaegar. And I think magic and her romance with Rhaegar represents Lyanna being true to herself, true to her differences, not fitting herself in this box. The life she and Rhaegar lead in the Tower of Joy is initially one where they've rejected noble families and conflicts and are living more "freely". Their bond is about non-conformity, in ways that just don't feel much like Sansa at al, who initially wants to be queen with Joffrey before the illusion is shattered? Obviously after Brandon & Rickard die, the tone changes with Lyanna's romance - it leads Lyanna on a grief-fuelled prophecy obsession and leads her to conceiving Jon, but that reaction feels quite unlike either of the Stark sisters.
In truth, while there are key similarities, I don't think Lyanna is entirely like Arya either. I would actually argue she's most like Bran - Bran who also declares 'I want to be a knight', Bran who also pursues magic instead of normalcy, a choice that is a selfish one, that costs lives. His reaction to the loss of his mobility and even loss of family is just to pursue magic more deeply despite the dark and dangerous path it provides. But I do think the Lyanna-Sansa comparisons often just reduce things down to Lyanna having a whirlwind romance that Sansa would fantasise about, but in text, ultimately becomes less rosy-eyed about and more realistic (while keeping some of her committment to kindness and idealism?). It just feels like a misread based on the fact that romance is fairly central to the character arcs, but they're both exploring different things with romance and about romance as a gejre.
Last thing, I do need to say this: I think it's ok for the characters to not be alike and Sansa is no less a Stark if she is not like her aunt.