As some of the other posters have pointed out, modern, high(er) end LCR meters essentially measure the phase angle between voltage and current and calculate L, C, R, and related quantities from that phase shift.
A lot of the differentiation between LCR meters is in things like measurement frequency, measurement accuracy, bias, and additional functions like frequency sweeps, transformer measurements, binning, etc., as well as fixtures, interface, and ease of use.
It's really hard to go back to using a handheld LCR meter (and I've used most of them) after using a benchtop LCR meter