So I've been thinking A Lot about Rouge and Amy recently (obviously lol) and I think I can finally put into words why I think they work so well together and why I'm so obsessed with them and their dynamic
Amy is loud, energetic, bubbly, and screams her love from the rooftops. She's never afraid to express her emotions openly, and that absolutely also applies to how she expresses love. Amy is as bold and bright as they come, a total open book, and an absolute sweetheart. However, she also struggles a lot with her own sense of identity and self respect. She's spent so much of her life chasing Sonic and trying to bend herself around him that she's sometimes unsure of who she truly is. Amy, whether consciously or not, hides her fears, identity issues, and lack of self respect behind her bubbly persona
Meanwhile, Rouge is a lot more subtle with how she expresses her emotions. Usually she expresses her love and care for others through actions, and it's more rare that she gets very sappy or emotional with her words. She's caring, but she's more subtle and quiet about it. Despite Rouge being more openly flirty and teasing though, she seems to bury a lot of her deeper emotions, whether that be due to fear or pride, it still remains true. She seems to downplay or outright ignore her own feelings much of the time, and she likely had a lot of emotions buried deep down that she hasn't allowed herself to express, or possibly even really acknowledge and feel even privately. Rouge, whether consciously or not, hides her deeper emotions and trauma behind her flirty and teasing femme fatale persona
When you put both of these people together, you get a dynamic that allows for mutual growth and healing. Rouge and Amy both have their own demons, but they can uplift each other with their strengths. Amy has someone other than Sonic who sees her for who she truly is, who loves her and wants her to see just how wonderful she truly is, and who is able to help her build her own sense of identity outside of the one person she had built herself around for so long. Likewise, Rouge has someone who she feels safe enough to be open and honest with, someone who encourages her to let herself be more vulnerable and open about how she's really feeling underneath the exaggerated flirtiness and teasing
Rouge is overall confident and self assured, Amy is overall deeply emotional and compassionate. Rouge represses her emotions, Amy struggles with her sense of self. Their flaws and their strengths perfectly sync with each other, both of them being able to provide exactly what the other needs, while also challenging them and encouraging them to grow and change
TL;DR: Rouge and Amy have an absolutely phenomenal dynamic in which they both perfectly balance each other out and compliment each other. Rouge gets the emotional support and encouragement to be more open and vulnerable from someone with limitless kindess and understanding, and Amy gets someone who isn't Sonic that sees her for who she truly is and can help her learn to truly value herself and build a stable sense of identity