No, the current passes through the coil, and the coil has no physical contact with the vehicle.

Look up inductive coupling. The basic idea is that a changing (AC) current in a conductor generates a changing magnetic field (Ampere's law). This changing magnetic field then induces a voltage in the second conductor (Faraday's law). This is the principle behind how transformers work.

As far as I understand, it's just electromagnetic waves. No current passing through to the car, but the coil can "register" a change in its magnetic field and can determine that it's a car and how fast it goes

Can confirm. Also interesting to note that motorcycles often have trouble triggering these sensors (a common trick is to stick a heavy duty magnet underneath).