"Grow up!" simply means act mature, responsible, and civilized.
The term "real man" is something most often used by women to shame their men into behaving the way (they) believe a man should behave.
The reality is there is no such thing as a "fake man" unless you discover a woman is posing as a man. The definition of "real man" will vary from woman to woman because each has their own (expectations) of men.
In modern times a lot of effort has gone into removing several gender traditions/stereotypes in order to push for equality for women.
Historically men worked outside of the home and women stayed home to care for the children, cook, laundry, and clean. The man did the yard work, barbequed, made sure the cars were serviced, killed any bugs that came into the house, and made minor repairs around the home....etc
Whereas many men today may take care of these things by making a few phone calls and hiring someone to do most of these duties. This shouldn't mean he is not a "real man". It just means he's not a man from the 1950s!
Today women have careers, may not cook, and can legally choose not to have children. And yet no one would say they're not "real women".
Several years ago a slogan was born: "Real women have curves". As if to say any woman who is naturally thin is not a "real woman". That's insane!
Once again it's an example of one group trying to make themselves feel better by putting down another group.
A "real man" is a man who will do whatever (that woman) believes men should do. It's a term intended to insult a man who isn't doing what she wants.
As for mature responsible adults; both genders should fulfill their financial obligations to creditors, respect and nurture their spouse, look out for their children's best interest and plan for the future.
Everything else is what makes both men and women "individuals". Some men are comfortable with crying and openly being emotional. Some women are not into flowers or romantic comedies. Whenever the term "real" is used in front of a gender it most likely harkens back to an era of (stereotypes).