What are the units of the product of two signals?
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.