Standard robot grippers are not designed for elasticity estimation. In this work, a professional biaxial compression device was used as a control setup to study the accuracy with which material properties can be estimated by two standard parallel jaw grippers and a force/torque sensor mounted at the robot wrist. Using three sets of deformable objects, different parameters were varied to observe their effect on measuring material characteristics: (1) repeated compression cycles, (2) compression speed, and (3) the surface area of the gripper jaws. Gripper effort versus position curves were obtained and transformed into stress/strain curves. The modulus of elasticity was estimated at different strain points. Viscoelasticity was assessed using the energy absorbed in a compression/decompression cycle, the Kelvin-Voigt, and Hunt-Crossley models. Our results can be summarized as follows: (1) better results were obtained with slower compression speeds, while additional compression cycles or surface area did not improve estimation; (2) the robot grippers, even after calibration, were found to have a limited capability of delivering accurate estimates of absolute values of Young's modulus and viscoelasticity; (3) relative ordering of material characteristics was largely consistent across different grippers; (4) despite the nonlinear characteristics of deformable objects, fitting linear stress/strain approximations led to more stable results than local estimates of Young's modulus; (5) to assess viscoelasticity, the Hunt-Crossley model worked best. Finally, we show that a two-dimensional space representing elasticity and viscoelasticity estimates is advantageous for the discrimination of deformable objects. A single-grasp, online, classification and sorting of such objects is thus possible. An additional contribution is the dataset and data processing codes that we make publicly available.