i think my big problem with people using "tomboy" as a descriptor for arya is that there's nothing wrong with the term as it is, it's a word used to describe a girl or woman who likes to take part in traditionally "masculine" activities and in later years has phased out of use as we learn more about gender identity and gender expression and what "masculine" and "feminine" even mean. it's just a word and by itself it might even apply to her just going by definition alone.
but that's not the issue i have with it, my problem with people calling arya a "tomboy" is that they're using it to justify why she has to be ugly and can never be a lady or a queen, when that's utter bullshit. they use the fact that she wears trousers for riding, and practicing water dance (which is only practical, who the hell is exercising in silk dresses?), and while hiding to say she despises dresses and femininity when that's just not supported by the text.
the only times she's ever complained about having to wear a dress were when she refused to go with sansa to the queen's wheelhouse in AGoT, during her water dance lessons because that was just impractical and when she's made to wear dresses and comb her hair and mind her courtesies once people found out she was a lady back in the riverlands and started putting all these expectations on her they didn't have a couple days before; and she even got sad when she ruined the acorn dress so it's not like she did it out of spite. other than that she never minded any of the dresses she wore in braavos, even wore some of them out of her own volition.
arya never once refuses to take part in household activities even if she's mostly doing them as a servant and out of survival, and we know she doesn't think any less of the women who do such things because she enjoys being around the small folk and the servants and even prostitutes and talking to them. arya stark loves flowers and people and naming babies and playing with them. she takes care of a baby herself while she's nothing but a child because she refused to leave her behind. arya is a romantic and nurturing and insecure about her looks because she wants to be pretty too. i'm not saying these are all things that define womanhood, but these are the traits that define hers.
the term is outdated and sexist and used to put women down and devoid them of their womanhood for daring to live out of the norm and what's expected of them by implying that by rejecting certain aspects of their expected roles they're trying to be like boys, and it's used by people who can't accept that sansa's own femininity is flawed and rooted in classist and patriarchal ideals of what a woman should be, and that both arya and sansa can be beautiful and feminine in two different ways. they're trying so hard to seem woke and supportive of this arya that doesn't exist who hates women and everything associated with them that they've circled all the way back to misogyny.