She probably designed it again when she got older to make it look better.

Soo I have noticed something interesting while looking for some screenshots, and I realized that this dragon design that Ryuuko has on the back of her jacket is basically still the same of the one she wears when she was a little kid.

I think it makes sense that she designed it herself especially because “Ryuu” (流) not only means “flow” or “style” but is also homonymic with “Dragon” (龍).