That protocol is called SQI (resembling SPI) meaning Serial Quad I/O Protocol.
You can sed the adress/command/data trough 4 pins instead of only one (SPI) and receive de data trough the same 4 pins.

Being half or full duplex is a characteristic of a comunication system regardless of the protocol you decide to use (of course some protocols allow full-duplex and some just don't).

Indeed, Dual and Quad I/O are not the same as full-duplex, they are just different protocols you can use (sometimes with the same device). Starting with SPI (I'm assuming that you know what this is) that has 1 pin to send and then receive (meaning that the protocol is half-plex); we then have Dual I/O that uses 2 pins to send and the receive (half-duplex), and finally Quad I/O that uses 4 pins to send and then receive (also half-duplex).