On a fretted electric bass the string vibrates between the metal fret and the metal bridge piece, and that vibration is transmitted magnetically to the pickup. Maybe I’m missing something, but I cannot logically see how the wood of the fingerboard could have an effect on the sound in that chain.
Fretless could be a different story altogether.
The strings vibrate on the resonator (neck / body) which affect the overtones and harmonics that the strings emanate. The transducers pick up these vibrations while also coloring the sound.
Fun Fact
There's also a difference between round woumd strings and flat wounds ;-)