You mentioned there is a mechanical analogy to resonance- does that mean that inductors have a resonant frequency that is a property of the material?

Yes, with the spring mechanical analogy to the $RLC$ circuit, an applied sinusoidal force is analogous to an oscillating power supply, and you will get a resonance for certain frequencies. However the resonant frequency is a property of the whole circuit, rather than just the inductor. We will discuss this further soon.