No, pickups don't function like microphones; they create an electrical signal in the pickup windings caused by the steel strings moving across a magnetic field, and in doing so also pick up the harmonic frequencies fed back to the string from resonant vibrations created within the bass - usually made of wood. As musicians we perceive these extra harmonics as "character", and it's why electric guitars and basses sound like guitars and basses, not synthesisers. Piezo transducers "hear" the sound one way, pickups do so in another.