People love to point out that Daenerys and Arya both take masculine actions and that this makes them unfit to be a female role model. Like one PFFFFT. And two, way to play into gender roles. I see where your feminism is at.
The thing is, in the series, both are proud to be women.
“The woman is important too!” Arya protested. (AGOT, p. 61)
“Woman?” [Daenerys] chuckled. “Is that meant to insult me?” (ASOS, p. 576)
Arya repeatedly asserts her gender throughout the series. She objects to the term Lady, as it refers to the profession of being a high-born girl in her society, not the word “lady” as a descriptor of gender.
Dany wears the title of Khaleesi proudly. “Mother of Dragons” proudly. When her people dub her “mhysa” she considers it incredibly moving and important. Her fondness for motherhood is incredibly interesting when one considers that often the only purpose of a woman in a narrative is to provide children. It’s even more nuanced in Dany’s case, since she was rendered barren at the end of the first book. She finds the ability to give comfort and care to others, and despite her inability to produce children, she considers being a mother to be the highest compliment of all.
And Daenerys doesn’t play the role of a patient, reactive mother. She knows what dangers are inherent in being a girl and she is incredibly aware of the misogyny of the society that she lives in. She even plays on it to give herself an advantage. “I am but a young girl and know little of the ways of war,” she tells people. As the audience we know that Dany has led an army, conquered three cities, and ruled over one. But because she is a small, young girl, people still underestimate and devalue her.
“Do you think I have forgotten how it felt to be afraid?” Dany asks, when challenged. She never forgets what it felt like to be a powerless girl in a whole of powerful men, and so she respects other women greatly. Dany has very close, meaningful relationships with her handmaidens. She confides in and often takes advice from Missandei, a former slave, who is still a young girl. When the Dothraki raid a village, to get supplies and resources for Dany’s quest to Westeros, Dany is appalled by the rapes of the village women. She takes all the women under her personal protection and forbids that any of the Dothraki touch them.
Similarly, that Arya quote up above? It’s from her very first POV chapter. She continues to respect women in her actions too. She is very protective over a small girl, Weasel, and is determined not to leave her behind just because she is small, slower than the rest of the group, and a girl. She befriends the women of a whorehouse, who society would deem the lowest of the low. Arya never shames them for their profession or position, she knows that it’s more important that they are kind to her. All of Arya’s personal heroes are women; she wants to be more like the fictional queens from mythology, and she names her direwolf after one of them.
Their support for other women is what makes Dany and Arya so special. They don’t judge women on looks or age or social status. They fight for the protection and rights of all the woman and girls they can.