It affects them quite severely, actually. In a study that's now 10 years old or older, children living with obesity rate the quality of their lives lower than children who are undergoing chemotherapy for cancer.
Stigma against obesity—bias against obesity—is one of the last publicly acceptable prejudices. That status of being overweight or obese is erroneously linked with gluttony, sloth and self-indulgence, whereas in fact it's a confluence of genetics and the obesogenic environment we have.
In my practice, I have many children who have cried when they're placed on the scale just so I can get their BMI. They are extremely self-conscious about their weight. They're getting teased in class relentlessly. This sort of state is linked with depression and anxiety in children.