You need to find a schematic for your car’s electronics in order to get a proper solution for what you’re trying to do. I would be particularly cautious about connecting a relay coil to some unknown wiring. The back e.m.f. from a discharging relay coil could easily fry a microcontroller if it happens to be connected to one.

The FM tuner is a narrow band pass filter centered at the carrier frequency - it doesn’t try to track the modulation. The FM demodulator then uses a VCO (voltage controlled oscillator) with a phase detector in a feedback loop, so that the output of the VCO tracks the modulated input signal. The input to the VCO then represents the baseband (demodulated) signal.