In a recent letter [N. V. Krishnendu et al., Phys. Rev. Lett. 119, 091101 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.091101] we explored the possibility of probing the binary black hole nature of coalescing compact binaries, by measuring their spin-induced multipole moments, observed in advanced LIGO detectors. Coefficients characterizing the spin-induced multipole moments of Kerr black holes are predicted by the "no-hair" conjecture and appear in the gravitational waveforms through quadratic and higher order spin interactions and hence can be directly measured from gravitational wave observations. By employing a nonprecessing post-Newtonian (PN) waveform model, we assess the capabilities of the third-generation gravitational wave interferometers such as Cosmic Explorer and Einstein Telescope in carrying out such measurements and use them to test the binary black hole nature of observed binaries. In this paper, we extend the investigations of [N. V. Krishnendu et al., Phys. Rev. Lett. 119, 091101 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.091101], limited to measuring the binary's spin-induced quadrupole moment using their observation in second generation detectors, by proposing to measure (a) spin-induced quadrupole effects using third generation detectors, (b) simultaneous measurements of spin-induced quadrupole and octupole effects, again in the context of the third-generation detectors. We study the accuracy of these measurements as a function of total mass, mass ratio, spin magnitudes, and spin alignments. Further, we consider two different binary black hole populations, as proxies of the population that will be observed by the third generation detectors, and obtain the resulting distribution of the spin-induced quadrupole coefficient. This helps us assess how common are those cases where this test would provide very stringent constraints on the black hole nature. These error bars provide us upper limits on the values of the coefficients that characterize the spin-induced multipoles. We find that, using third-generation detectors the symmetric combination of coefficients associated with the spin-induced quadrupole moment of each binary component may be constrained to a value ≤1.1 while a similar combination of coefficients for spin-induced octupole moment may be constrained to ≤2, where both combinations take the value of 1 for a binary black hole system. These estimates suggest that third-generation detectors can accurately constrain the first four multipole moments of the compact objects (mass, spin, quadrupole, and octupole) facilitating a thorough probe of their black hole nature. © 2019 American Physical Society.