Both are, or were, magic users in a sea of nonmagic users.
Both have been locked away through no fault of their own (arguably Elsa’s striking Anna counts as having done harm, but not intentionally, nor does it justify a lifetime of isolation).
Both are naive as a result but are still intelligent (Rapunzel with her analysis of star movements, Elsa with her construction of a complicated ice castle).
Both know what it's like to give up their freedom for the sake of keeping someone they care about alive (though Rapunzel’s was only threatened I have no doubt she would keep her word, and she's already seen the world and knows what the difference between it and being locked away are).
Both are artistically inclined.
Both are royals who love their people (even if Elsa fears them, you can see the happiness in her face when she looks around at her subjects at the end).
Both are sweet and loving when they're allowed to be (Rapunzel when she interacts with people, Elsa when she's not obsessing over her fear and guilt).
They're also opposites in a lot of ways.
Elsa is pessimistic, dwells on her bad emotions, doesn't know how to deal with her anxiety besides keeping it locked inside.
Rapunzel is optimistic, brushes aside most bad thoughts and goes to other people and relies on them when she's feeling poorly.
Elsa thinks of her power as a sign she's a monster until she is shown she's loved.
Rapunzel thinks of her power as a gift that is reason enough to keep her locked away.
Elsa bears the burden of royalty, responsibility and it, in conjunction with hiding her power, is killing her.
Rapunzel does have responsibilities but for the most part it's the boredom that's killing her.
Elsa is broken, but healing.
Rapunzel, whether she has magic or no, is at heart a healer.
They share many crucial traits in common but the places where they differ could make for a fascinating relationship.
