Radio receivers used to work (and I believe still do) on the heterodyne principle at the tuner end. This means the system to tune a particular frequency first generates a frequency signal just slightly different from the signal desired. (Today's PLL oscillators are much better in many ways than the older types.) This is then mixed with the whole mishmash of incoming signals picked up by the antenna and amplified by a low-noise broad-band stage. It is then run through a series of narrow passband filters and amplifiers tuned to a specific centre frequency. (Again, today's designs use quartz crystals to achieve very narrow passbands with sharp cutoff cliffs.) This section is called the "IF strip" (IF = Intermodulation Frequency), and it is tuned to the exact difference between the broadcast frequency and the receiver's internal tuning oscillator. Thus only the radio's very first stage, an amplifier, needs to be high gain and wide bandwidth. From there on the amplification is done on a narrow bandwidth at a fixed centre frequency, allowing very high gain with low noise. After this is done, the resulting strong modulated IF carrier is processed with the appropriate type of detector to extract the original audio signal that was broadcast.
So OP's basic question is a reasonable first thought - if the local oscillator at the front end MUST be different from the broadcaster's frequency by some exact and known amount, you can determine what station is being listened to. Wall Street and Potted Meat pointed out the flaws in the logic - the frequency determination would have to be VERY exact, and the signal generated by the internal oscillator should NEVER be leaked out of the receiver so an outside antenna could pick it up. Even if you could mount a pickup coil around the car's antenna (which is certainly NOT the subtle detection system OP was dreaming of!), I doubt the signal from the local oscillator would be strong enough to detect and analyze.