Can a single Arduino appear as more than one USB device?

I want to, eventually, after a lot of work, convert my NES Advantage controller to USB. I own a retrolink NES to USB adapter, and it works fine.

Here's the problem: The Advantage has a switch to go between Player 1 and Player 2. I originally thought of buying a second USB adapter and plugging them both into my MacBook, but one of the apps I use won't recognize one of them if I have two identical game controllers plugged in.

So, if I have two USB devices that identify as "NES Controller USB" to the computer, the app (MidiHID) will only be able to use one of them. I'd like to be able to switch between P1 and P2 to control different functions.

My goal here would be to connect both outputs from the Advantage up to a single Arduino, connect a USB cable, and have my computer see two devices such as "Advantage USB P1" and "Advantage USB P2" so I retain the functionality as a two-player game controller. Is this possible, or would I have to use two separate Arduino boards?