What are the units of the product of two signals? In a standard mixer, we multiply two signals $x_1(t)$ and $x_2(t)$, each with units of volts $(V)$, and get at the output:

$$y(t)=x_1(t)\,\cdot\,x_2(t)$$

I have always thought that the output of mixing operation was another voltage signal. But how can this be? Looking at the units of the factors, we multiply a quantity of volts, $x_1(t)$, by another quantity of volts, $x_2(t)$. So shouldn't the units of $y(t)$ be $V\cdot V=V^2=W$?
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.