Does FFT magnitude of recorded sound depends on the bit depth used?
With very few exceptions any audio processing these days is done in floating point, i.e. the data that comes through the ADC is immediately converted to floating point numbers and normalized so that the max voltage corresponds to a value of 1.0. This eliminates any dependency of the scaling to the number of bits.

Fixed point processing is quite difficult and typically requires specialized data types and intensive scaling and noise management. 

The output of an 8-bit ADC could be interpreted as the numbers $-128,-127, ...+127$ but it's almost always better to think of it as $-1, -127/128, ..  -1/128 , 0 ,+1/128 ... +127/128$