I don't know what it's called in English, but it's called a 'Shampoo Hat' in Japanese, and it's very much for children; as it stops the shampoo from getting in your eyes, it's for those who don't have the ability to wash their hair and keep their eyes shut at the same time.
The only time you'd see an adult with them is the disabled or the elderly.
So here, it's saying that Kongou is more childish than Akatsuki, going by the trembling and the SFX by the eye implying that she's having a hard time keeping her eyes closed.
Oh? That's awesome, Akatsuki!
Washing your hair all by yourself♪D-Don't you be treating me like a child! Even I...!!I see♪
But, if you say 'Lady', you'd be talking about Kongou, right?O-Of course I am - I'm a first-class Lady, after all!