In haptic object discrimination, the effect of gripper embodiment, action parameters, and sensory channels has not been systematically studied. We used two anthropomorphic hands and two two-finger grippers to grasp two sets of deformable objects. On the object classification task, we found: 1) among classifiers, SVM on sensory features and LSTM on raw time series performed best across all grippers; 2) faster compression speeds degraded performance; 3) generalization to different grasping configurations was limited; transfer to different compression speeds worked well for the Barrett Hand only. Visualization of the feature spaces using PCA showed that gripper morphology and action parameters were the main source of variance, making generalization across embodiment or grip configurations very difficult. On the highly challenging dataset consisting of polyurethane foams alone, only the Barrett Hand achieved excellent performance. Tactile sensors can thus provide a key advantage even if recognition is based on stiffness rather than shape. The dataset with 24 000 measurements is publicly available.