Can a modern multimeter be classified as a discrete component tester?
I think 'discrete component tester' may be an alternative name for an 'LCR meter', which specializes in measuring resistance, capacitance, inductance, and related factors (D, Q, ESR, etc.).  These devices are precision instruments which don't passively measure, but actively apply test signals and measure the response.  They will also generally have features to increase accuracy, including kelvin connections (separate source and measurement leads) and guard terminals (driven output that is used to minimize the effects of leakage).  Some multimeters may have some of these features, but generally a dedicated LCR meter is more accurate and can measure more than one parameter at the same time (C and D or ESR, L and Q, etc.).