Raspberry Pi, SPI, multiple data in lines
The datasheet passage you quoted answers your question.  The A/D can be used to get both readings simultaneously.  Some SPI implementations allows for multiple input lines, sometimes bi-directional.  If the hardware doesn't have this capability, you could implement it in firmware-only.

Alternatively, you can just ignore the second data line.  The first reading is sent in the first 16 bits, then the second in the next 16 bits.  This is compatible with standard and common SPI hardware, but takes more clocks to get the same data.