There ought to be many protections built in the car by design and the charger's surge dump rating if it complies with automotive standards, which might not be the case if your power connection is not switched off during start sequence such as headlamps and in some cases dash power, heater motors etc are.
*WHat is a surge dump?** ANy inductive load such as releasing a starter solenoid or an inductive starter motor or an AC clutch returns some moving stored energy back to the battery. The spike or rise in voltage depends on the battery condition or specific gravity s.g. and its internal resistance ESR which are inversely related. low s.g means high ESR. and poor Ah rating. But it also means higher voltage drops and spike rises when the load is switched off.
For this reason and also to reduce the load , historically cars have used a built in accessory switch to OFF just before and after Starter switch is engaged. But if your power source is not then your chargers must be rated for the standard automotive load dump rating as all accesories and peripherals must. The start load dump is the highest so that is why accessories are disabled during start sequence. The main load dump causes are AC clutch and battery booster errors from reverse connections and the energy stored in the inductance.
Other info
Batteries all have an effective series resistance ESR which obeys Ohm's Law. This is why Cold Cranking amps is rated at 7.5V so the maximum current a battery can deliver at cold temp is determined by 12.5V-7.5V or a maximum drop of 5V/ESR of the battery. In reality this never occurs unless the motor is almost seized from ice on the moving parts so the voltage drop and cranking current is less.
But assuming your condition is less severe, I suggest it is because your phone chargers are intended for home use and not automotive use for the surge is not absorbed by an MOV or the overvoltage protection is too slow for the rise time of the spike. EMF or Vrise = L*dI/dt so both L and I affect dt the rise time of the spike. Then battery ESR * dI affects the rise of that voltage spike.
DC-DC converts used in phone chargers have many other protection mechanism such as low pass filters and high speed regulators but sometimes they over react to a dip then when the voltage returns it is pumping out too much voltage.
Next the phone is responsible for drawing the actual charge current it needs regardless of the voltage but that too has limits so they often include surge dumping MOV's.
Conclusion
Without knowing your details of switched accessory, battery condition , load rating and phone charger automotive rating and phone surge protection, it's hard to isolate the fault, but I made my best guess.
