Do you really think she thinks of you as a parental figure, at 21? And if not, do you think perhaps you shouldn't think of yourself as having a parental role in her life? Perhaps if you reframed the relationship in your head, from being step parent and step daughter, to your partner's partner and her child, it might be easier for you to detach, and enable them just to get on with it, in their own time and way?
If the parents have concerns, then they need to be working with their daughter and finding tools, support and solutions.
I think you (to a certain extent) mean well, but I'm not sure your involvement is necessarily helping. And I do think perhaps that you and she are not a good fit personality wise, perhaps rub each other very much up the wrong way, and you need to be a bit more honest with yourself about your attitude towards your partner's daughter, and your motivations for wishing her to 'move on'.
She's 21. Many 21 year olds are very immature, particularly if they've been babied and done for, most of their lives. Many are still heavily dependent on their parents. Many of them are introverts, far more into their phone and the internet than partying. Many of them are heavily emotional and overly dramatic. Many of them can't stand their parents, let alone their parent's partner. Many of them are as annoying as all get out - thinking they know it all and should be treated like fully functioning adults when they're nowhere near behaving like fully functioning adults! It sounds like this is the model your partner has at this point in time. . .
But basic groundrules and expectations for you and your partner to set for everybody in the home you own together is fair enough. Basic standards of tidiness, cleanliness, everyone contributing to the running of the household, and basic standards of courtesy towards each other (including not nit-picking, being hyper-critical, or making it obvious how much someone is irritating you by their manner of speech, not throwing emotional outbursts when someone upsets you, but walking away - maybe shutting yourself in your room and going online so as not to have an emotional outburst. . .)