The single gate MOSFET is a three electrode device, or triode, whereas the dual gate MOSFET is a four electrode device, or tetrode. When the two gates are strapped together, we turn the tetrode into a triode. The former has a transfer characteristic which is S-shaped, so we see flattening of the peaks, and above a certain threshold we observe odd-order harmonics and intermodulation. On the other hand, the triode has a square law characteristic, so for large signals we see even-order distortion, giving rise to even harmonics. This effect is linearised to some extent by using a high value load resistance.
It is likely that generation of even order harmonics will be less troublesome than the generation of intermodulation products, which can lie near the wanted signal and be impossible to remove. A disadvantage of the triode is the higher capacitance between drain and gate, which is multiplied by the gain and appears across the input (the Miller effect). In the present application the device is driven by a small antenna element having a capacitance of a few picofarads, so the input capacitance needs to be somewhat smaller than this.
The two gates of a dual gate MOSFET each control the drain current, so the drain current is proportional to their product. As you mention, this a enables the device to work as a multiplicative mixer, where the signal V is applied to G1 and the local oscillator (LO) to G2. The instantaneous LO voltage then controls the gain applied to the signal, resulting in mixing action.
When G1 and G2 are strapped together, as in the present case, the drain current will depend on Vg1 x Vg2 = V^2, so we obtain a triode characteristic.
Although it is my understanding that, with a low resistance load, FETs exhibit a square law characteristic, for a vacuum triode I believe we see an exponent of 3/2.
