Neural audio codecs are increasingly used for low-bitrate speech coding and serve as tokenizers for speech language models, yet it remains unclear how their codebook hierarchies encode speaker traits and speech characteristics. We systematically study 17 multi-codebook neural audio codecs across seven traits: sex, age, accent, voice quality, emotion, speech flow, and non-verbal sounds. For each codec, we reconstruct speech using partial codebooks and use machine listeners to compare trait predictions on each reconstruction. We measure prediction agreement with reweighted Cohen's \k{appa}, and define trait depth as a proxy for the codebook depth by which each trait is encoded and recoverable. Overall, sex has the shallowest trait depth, followed by speech flow, whereas voice quality and age group require deeper stacks; emotion, accent, and non-verbal sounds fall in between. Within each trait, speech-flow classes are at relatively similar trait depths, whereas emotion, accent, and non-verbal sounds differ more. Additionally, we explore how certain codec design choices relate to trait depths. These findings show that codebook hierarchies are trait-dependent and may not exhibit uniform coarse-to-fine structures, motivating trait-specific codec evaluation and design for future applications.

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