Why to measure phase and amplitude, and not just an amplitude ?
The peculiarity comes with interpretation of word "Impedance".
The best way to avoid confusion is to say "Complex Impedance" and "Impedance Magnitude". For components, it is best to refer to "Reactance" and "Resistance" separately. The Siemenses can be treated the same way: "Complex Admittance", "Admittance Magnitude", active and reactive components "Conductance" and "Susceptance". And characteristics of Impedance and Admittance are blanketed with word "Immitance". I'd rather avoid this word completely.
Why is there 2 numbers needed instead of just one ? The reason is that real life inductors and capacitors are not fully described with single number. There is parasitic resistive component in addition to desired reactance value. This parasitic component can be measured as second number. And since Complex Impedance is a vector sum R+X, the parasitic component R is expressed on schematics as included "in series" to ideal X of L or C component. The R is often named ESR - Equivalent Series Resistance.
The ESR is just an approximation, attempt to describe some real part with 2 elements: reactive X and resistive R. ESR is the name of a Model of theoretical part, which is fully described with 2 numbers. This 2 numbers is just another approximation (better than single number, but still incomplete).
For further detailing of equivalent network of real life parts, experimenter will need the "Vector Network Analyzer" VNA which is capable of describing the network of more than 2 components. Network Analyzers use more measurement points at different frequencies (still gathering a single pair of numbers at each frequency point). LCR/ESR meters like UA315 may or may not use multiple frequencies to fit data into 2 equivalent component model. The only difference is that LCR/ESR meters are focused on single or 2 component model.
To line this up:
And more: If LCR/ESR meter produces data from multiple frequency points, will it allow to have say 3 component model ? Yes, it is completely possible. The only significant difference between LCR/ESR meters and sophisticated Network Analyzers is software. And in general, there is no need for Analyzer to analyze (do analytical math to derive equivalent network) anything. The simple example is devices charting the Frequency Response Charts for speakers. It is the user, not the software who interpret the chart.
In next parts of the story, this 2 numbers will be used to express 2 component model. Any more sophisticated cases are up to the user, who will decide to redesign the firmware.