first of all you need to make sure your girlfriend is okay with you being a "parent" figure to her son, which I am assuming you want to be since you said "we are the adult", you guys may have establish that already but it is not clear in the question.
Then, you need to talk to her directly. It is probably the way she will be less defensive about her parenting skills. Let her know you are tying to help make her life easier and she needs to show her son she is the adult. He will not respect you as an adult (assuming you recently came into is life and not being there to raise him) until he respects her.
I think that if you confront her then she may get very defensive, especially because it is in regards to her son. What you could do is in casual conversation suggest that there are some parenting classes that would be good for both of you to take together. Just make sure that you have picked ones that address the topic.
I'm not sure what you are asking actually. "How can I make my girl see that" all makes sense, but... "we are the adult not her son" is gibberish. Are you asking for advice on how to tell your girlfriend that you are an adult, and not to be treated like a kid? Or are you asking about her son?
Don't MAKE your kid KNOW you're the adult act like an adult and she'll understand . Life is way to short to argue fuss and fight . U argue they will. U make them whatever they rebel . Raised 3 kids by myself ! All above 30 now and nice people . They fight with you go out for ice cream and talk or out to eat just u-2 . No fighting or everyone suffers the consequences .