Note that the frequency won't change if you put it into an inductor: the frequency is set by the ESP32 and as such isn't something the inductor would change. (For reference: a fair few of the 555-based circuits use the loop as part of the circuit that sets the frequency, and as such the frequency would change there.
One way I can think of doing things is to use the inductance as part of a filter. You use the inductor (=the vehicle loop) combined with a capacitor to filter the signal generated by the ESP32, then have some kind of peak detector to detect the amplitude of the resulting signal. If the inductor value changes because a car passes over it, the filter de-tunes and you should see a change in amplitude at the peak detector.
If you want to play around with that idea, I built a simulation
here. Change the inductance value by a little bit (e.g. from 100uH to 99uH) and the output of the peak detector changes by a lot. Note that this circuit may actually be *too* sensitive; you may need to work around this in software by writing some code to detect at what frequency the maximum output is and use that frequency as an indication of the loop inductance. (Also note that this is entirely theoretical; I have no experience with loop detectors and have no clue if this setup would work in practice.)
