Being a grown up as per my mom when I just turned into an adult, was to act your current age. More responsible, forget your child like ways, be more serious and leave those graphic tees behind.
To me, it meant turning into someone I was not, just when I had figured out who I was.
Now, at 35. My mom changed her views after seeing a different side of what it is to be a grown up. I see a change in her personality as I am very comfortable with who I am, even if I'm different from adults she knows. I always thought we were very different people, but as it turns out, she was just hiding that side of her all these years. I think it's because it was what she was told she had to be, to become grown up.
It doesn't mean you leave everything you are behind, it just means that you take responsibility of your life. You become independent, but know when to ask for help. It's to fall down and being able to pick yourself up to keep going. It's taking the consequences of your actions and do what needs to be done to survive, even if it's a job that you hate.You don't need to be an "adult" to be grown up. You can do that at any age.
But, it certainly doesn't mean you stop having fun and have to hide your true self to become like everybody else.