The canonical approach is an impedance bridge, with I-Q detection. That is, measuring the sin and cos components of the output, relative to the source (which might be defined as purely sin). This gives a complex result, i.e. including inductance and resistance.

Depending on the bridge design, I and Q may vary in a straightforward manner, or jointly, as inductance or resistance varies, but in any case there exists a transformation between the measured signals and the element's reactance and resistance. You could arrange a circuit to read mostly inductance, or impedance, but you might as well go for both, the added effort is small and the complete measurement makes more opportunities for measuring precision.

For more simpler analog approaches, you can arrange an oscillator to run at resonance (no need to track resonant peak), and count the frequency, and measure the current consumption, to find inductance and resistance respectively. Generally with more sources of error than the bridge approach. Or even just do a dumb impedance divider and assume voltage sensed corresponds to inductance -- it doesn't, it depends on impedance and phase -- but this gets close enough for most capacitor ESR tester circuits for example.

I'm particularly a fan of doing the I-Q (synchronous detector) method with an MCU, as it's pretty easy to do with a moderately fast ADC (for 60kHz, at least 240kSps is required; many MCUs offer this), and a bit of clever math.

Tim