In the town of Soquel, near Santa Cruz, CA, there is a Home Depot. In it, there are two bathrooms, one for "Men" and the other for "Family." The message is certainly not subtle.
Family means a father can safely take his daughter into the restroom or a mother can take her son in. Couches inside of restrooms are for women who need to attend to nursing their baby. These types of facilities are everywhere.
Family restrooms are for moms with small boys or dads with small girls. Chillax...
I think you might be mistaking the purpose for those two designations. Moms who are out shopping, and have little kids with them, want to be able to take them to the bathroom, if they, or she, should need to go. And we DON'T want to let our little boys go into the Men's bathroom all alone, whereas a Dad probably wouldn't be worried about about his daughter going into the Ladies' room alone. So having a 'Family' bathroom makes it easy for moms who have their little sons with them. The boys won't feel strange, because they're not going into the 'Ladies' room, they're going to the 'Family' bathroom.