Hey everyone, I have found the issue.
To connect the PS2 controller, I am using a "duo" USB adapter. This adapter allows for two controllers to be connected. Using jstest, I noticed that the first controller is assigned from the right port, and second from the left port. Originally, I connected my controller to the left. Once I switched to the right (and reconfigured via Recalbox) then N64 worked fine.
So, I'm not really sure what is causing this. I am guessing because of how mupen64plus works, it uses the identifier in the ini file (in this case "Twin USB Joystick") to match the controller, and it just matches the first one that it finds with this name. Because there are two, but the first isn't connected, that is what it is using.
I'm not sure if this bug can be fixed, but I hope so. If not, can I recommend putting something about this in the Recalbox wiki so that if people search they can easily find a solution? Here's some initial content:
Configure PlayStation DualShock 1 and 2 Controllers
Most emulators will work automatically.
However, for N64 there is one issue. If you are connecting your controller using a two-controller-port-to-USB adapter, then one of these ports will be considered "controller 1" and the other "controller 2". In one test, "controller 1" was assigned to the right port on the adapter. N64 emulation will expect you to be using "controller 1". If you are having issues, try connecting your controller to the other port and reconfigure it.