If the multiplier takes two voltages as input and returns a voltage as output, then there is necessarily a constant involved, with units of [1/V]. Take for example, AD633 (which was the first search result). The output is the product of the 2 inputs times a constant: $V_{out} = \frac{V_1 \times V_2}{10V}$ So the output units are Volts.


In addition to Juancho's answer for the general mixer, I would like to give an example for a more simpler frequency mixer most commonly used in communication systems to shift the frequency spectrum of a message signal up or down for transmission or reception etc. The simplest understanding of a physical realisation of a mixer assumes an on-off switching ...


Mathematically speaking $V^2$ is perfectly alright to use. Physically I am not sure you can multiply electrical signals like that.

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