tan delta testing is roughly the same as power factor testing (doble testing) of insulation.
It provides some indication of moisture, contamination, or insulation defects.
Also the powr factor can be measured at two voltages and a tip-up can be computed which indicates resistive losses increasing more than linearly with voltage. Possible indication of voids, but present to some extent on most motors.
Capacitance is usually determined in conjunction with the test, and phase-to-phase tests may be conducted in addition to phase to ground if neutral connection is accessible.
I don't believe this is used very often as a routine field test. The equipment is somewhat expensive.
It has been suggested in some Doble papers that this test is particularly useful as a quality-control test of windings prior to insulation. Note that after coil installation the test is somewhat limited for high-voltage motors with corona suppression system because the non-linear leakage current (increase faster than voltage) in grading/gradient treatment at slot exit cannot be effectively guarded out after installation.