Gimel Hooper, 49, who is Black, didn’t want to show “Frozen” to his daughter for similar reasons. But after she received an Elsa doll for her 4th birthday from a friend’s mother, it became his daughter’s favorite toy. “We couldn’t turn left or right without seeing ‘Frozen’ something,” said Mr. Hooper, whose daughter is now 10. “That song was constantly playing,” he added of “Let It Go.” “Every kid knew it.”

While Elsa sings in the dress, she appears to gain a sense of invincibility, and some parents described Elsa costumes as having a similar effect on their children.

Mr. Hemmig said that, at the peak of Jace’s “Frozen” obsession, he wore his Elsa dress to Target several times. Jace’s mother, Jean Hemmig, 45, said they let him wear the dress in public because “knowing we were confident in him” would help Jace feel even more confident.

Britta Shine bought a dress for her then-4-year-old foster daughter, who had experienced abuse and neglect, as a way for her to process the intense emotions she sometimes felt. It quickly became her favorite thing to wear. Ms. Shine, who lives in Detroit and has a Ph.D. in developmental psychology and infant mental health, said her foster daughter, now 6, stands up straighter when she’s wearing the dress. “She feels empowered by it,” Ms. Shine, 42, added.

Meghan and Kevin Dexter, 42 and 43, said that their son liked to put on an Elsa costume as a toddler and play with friends and family in their backyard in Richmond, Va. He never wanted to wear it elsewhere, but his parents would have let him. “If we had to modify every behavior that might get made fun of,” Ms. Dexter said, “we would be dictating way too much about our kids.”

Tammy Hart’s 5-year-old, Rocco, also started dressing up as Elsa as a toddler. Ms. Hart, 43, didn’t mind at all — until Rocco asked about wearing the dress to school in Brooklyn.