Speaker identity plays a significant role in human communication and is being increasingly used in societal applications, many through advances in machine learning. Speaker identity perception is an essential cognitive phenomenon that can be broadly reduced to two main tasks: recognizing a voice or discriminating between voices. Several studies have attempted to identify acoustic correlates of identity perception to pinpoint salient parameters for such a task. Unlike other communicative social signals, most efforts have yielded inefficacious conclusions. Furthermore, current neurocognitive models of voice identity processing consider the bases of perception as acoustic dimensions such as fundamental frequency, harmonics-to-noise ratio, and formant dispersion. However, these findings do not account for naturalistic speech and within-speaker variability. Representational spaces of current self-supervised models have shown significant performance in various speech-related tasks. In this work, we demonstrate that self-supervised representations from different families (e.g., generative, contrastive, and predictive models) are significantly better for speaker identification over acoustic representations. We also show that such a speaker identification task can be used to better understand the nature of acoustic information representation in different layers of these powerful networks. By evaluating speaker identification accuracy across acoustic, phonemic, prosodic, and linguistic variants, we report similarity between model performance and human identity perception. We further examine these similarities by juxtaposing the encoding spaces of models and humans and challenging the use of distance metrics as a proxy for speaker proximity. Lastly, we show that some models can predict brain responses in Auditory and Language regions during naturalistic stimuli.