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André Henriques
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If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .

The "quntum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $\mathbb R_{\ge 0}$-valued invariant with all the properties that you want.

See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.

André Henriques
  • 43.2k
  • 5
  • 130
  • 264