Harnessing the inherent anti-spoofing quality from electroencephalogram (EEG) signals has become a potential field of research in recent years. Although several studies have been conducted, still there are some vital challenges present in the deployment of EEG-based biometrics, which is stable and capable of handling the real-world scenario. One of the key challenges is the large signal variability of EEG when recorded on different days or sessions which impedes the performance of biometric systems significantly. To address this issue, a session invariant multimodal Self-organized Operational Neural Network (Self-ONN) based ensemble model combining EEG and keystroke dynamics is proposed in this paper. Our model is tested successfully on a large number of sessions (10 recording days) with many challenging noisy and variable environments for the identification and authentication tasks. In most of the previous studies, training and testing were performed either over a single recording session (same day) only or without ensuring appropriate splitting of the data on multiple recording days. Unlike those studies, in our work, we have rigorously split the data so that train and test sets do not share the data of the same recording day. The proposed multimodal Self-ONN based ensemble model has achieved identification accuracy of 98% in rigorous validation cases and outperformed the equivalent ensemble of deep CNN models. A novel Self-ONN Siamese network has also been proposed to measure the similarity of templates during the authentication task instead of the commonly used simple distance measure techniques. The multimodal Siamese network reduces the Equal Error Rate (EER) to 1.56% in rigorous authentication. The obtained results indicate that the proposed multimodal Self-ONN model can automatically extract session invariant unique non-linear features to identify and authenticate users with high accuracy.