What can I store inside the HORI Puff Pouch?

It is designed to transport and store your system and accessories, and the product overview says it has space for games, Joy-Con, and other accessories.

Is this carrying pouch compatible with both Nintendo Switch and Nintendo Switch 2?

Yes. The product description states that it is compatible with Nintendo Switch and Nintendo Switch 2, and the product overview also says it works with Nintendo Switch 2, Nintendo Switch, and Nintendo Switch OLED Model.