Our fans have the same design, but mine seems to be a little smaller and lighter than yours.
A group of folding fans, no matter how big it gets, is still a fan.
The reading for 団 is an exceptional reading. It's the same as 内's reading so you can think about how a group of folding fans to make a big fan is just a bunch of fans inside of fans.
The reading for 扇 is also an exceptional reading. Imagine a walrus (わ) using the 内 fan.