You'll find her in just about every series that features a club, a sport, an or art form - the girl who doesn't do the thing, but supports the boys and men in her life in their endeavors. She might be there to do emotional triage, make them food, and sometimes even serve as a love interest.
This archetype is exhausting because it positions women as inherently secondary to men. These characters don't have their own interests, they exist adjacent to men. It's sexist, it's bullshit, and it sucks.
Many of these characters are club managers for sports teams or other school clubs. While managing a club takes hard work and skills that shouldn't be discounted, the girl is rarely gung-ho about her role. Even when she is, she still occupies a support role. She's never the star, she's not trying to win or to improve herself. She's just sort of...there. At times, it seems like her only role is to have breasts. Because the straight male audience is tired of looking at muscley dudes all the time.
Sometimes, this girl has no defined role other than love interest. One notable example of this is Ritsuko Mukae from Kids on the Slope. The series focuses on the tumultuous relationship between Ritsuko's two best male friends, Sentaro and Kaoru, which is expressed through their mutual love of jazz.
Ritsuko doesn't play music, and neither do any of the other female characters in the show. Music is exclusively the provenance of the men. Ritsuko facilitates their musical achievements, while also smoothing over their emotional troubles and making peace between the two when they fight. She is asked to sing, once, and she does it well, but it's a one-off thing that never goes anywhere. She doesn't have the passion that Kaoru and Sentaro do, and it doesn't really matter to her or anyone else if she has a creative outlet. Her only real engagement with art is knitting - but knitting only matters to her because she's trying to make something for Kaoru, who she is in love with. It's not about the knitting.
Considering the fact that a Kids on the Slope is about emotional connection through music, leaving the main female character out of that theme is pretty glaring.
One particularly awesome subversion of this trope is Konatsu from Showa Genroku Rakugo Shinjuu. Konatsu loves and excels at rakugo, a form of Japanese storytelling. Unfortunately, she isn't allowed to make a career out of it - it's the 1970's and the profession is closed to women. She teaches Yotaro how to perform rakugo, and she teaches her own son how to do it, but it's not until the very end of the series that she's actually performing the art she's loved since childhood.
Konatsu is different from other characters in that she is driven by an inner passion. She cares about achieving something herself, not living vicariously through the men in her life. She's hemmed in by reality, but she's trying. Sometimes she gives in, sometimes she fights. That's what makes her human. That's what makes her a great female character.
Unfortunately, Konatsu is an exception rather than a rule. Most female characters hanging around male geniuses don't seem to be passionate about much besides ameliorating their male friend's issues.
All this said - Japanese feminism isn't American feminism. As an American, I'm in no position to say how these Japanese women should be represented in anime. All I can say is that this? Isn't working. Women are full human beings and they deserve more than being someone else's emotional life support.