Sports in general is a competitive activity. Every rec team is different, but for most cases, yes it's relaxed. At the 1st grade level, Refs don't call travel, double dribble, back court violation, etc. unless it's abused. The first few grade level is about learning and enjoying the game. There's always going to be an alpha (coach's daughter) that plays 90% of the time and usually the best player. Things change when you hit 4th/5th grade with winter travel/select teams. Usually all the best rec players try out for 10-20 roster spots (1 or 2 winter teams).

Your question is flawed. Girls sports can be just as competitive as boys. I am not sure where you're getting your premise.
Where I am from, 1st grade rec basketball is very laid back for both boys and girls.

My daughter found out some of her school friends did winter basketball, and she would love to join them this coming winter. They all will be first graders in the fall for rec. basketball. Boy rec basketball is really competitive and I don't really like much about the culture. How competitive is about girl rec. basketball for 6-7 years old this winter? It will be her first time doing sport team , and she does not know much about basketball. I wonder if girl sports overall are more relaxed or laid back?