  The measurement of multipole moments of astrophysical objects through
gravitational wave (GW) observations provides a novel way to distinguish black
holes from other astrophysical objects. This paper studies the gravitational
wave radiation from an extreme mass ratio inspiral (EMRI) system consisting of
a supermassive Kerr black hole (the primary object) and a spinning stellar-mass
compact object (the secondary object). The quadrupolar deformation induced by
the spin of the secondary is different for different astrophysical objects. We
compute the effect of the quadrupolar deformation on the GW phase and provide
an order of magnitude estimate of whether LISA can distinguish different
astrophysical objects through GW phase measurement. We find that although LISA
can not distinguish between a black hole and a neutron star, it can distinguish
black holes from a large variety of highly spinning astrophysical objects like
superspinars and highly deformable exotic compact objects like boson stars for
EMRI systems with relatively large mass ratio ($q\sim 10^{-4}$). Furthermore,
we show that the effect of spin-induced quadrupolar deformation on the GW phase
for white dwarf and brown dwarf-EMRI systems can be quite significant even for
small values of mass ratio ($q\lesssim 10^{-6}$).
