There she was more trying to become Batman's Robin after she debuted, which she basically was even if he didn't hand her a Batwave until the season 3...
I mean, I find anime a perfectly valid and more true-to-source way of bringing it to life in certain respects, and can better adapt the material, but...
I know, but she's still commonly seen or regarded as being a Batfamily member or sidekick, even if she's not as closely connected to Batman as Robin...