Based on our recently proposed shape manifold identification procedure, we investigate the identifiability of material properties with instrumented indentation test using diverse indenter shapes (conical/spherical) and simulation setups (prescribed force/displacement). We extend the protocol by taking into account both imprint and P-h curve, dissect the difficulties of material characterizing and re-stress the non-uniqueness of solution to an inverse identification. This work highlights the manifold approach in estimating the maximum number of independent material parameters that may be determined from an indentation model. The observations are verified by identification procedure using synthetic data over successive local manifolds defined in respective shape spaces.