This distance is the "obvious" singular value version of the class of Finsler metrics on the cone of positive definite matrices. A basic paper (not the earliest, but one with useful references that'll be handy) to start off is: [On the exponential metric increasing property] 1 by R. Bhatia.

Once I get back to my computer, I'll try tracking down some more info, especially, that with a longer historical perspective.

This distance is the "obvious" singular value version of the class of Finsler metrics on the cone of positive definite matrices. A basic paper (not the earliest, but one with useful references that'll be handy) to start off is: [On the exponential metric increasing property] 1 by R. Bhatia.

Once I get back to my computer, I'll try tracking down some more info.

EDIT. The three references cited below [1-3] will provide a deeper historical perspective. I am including a proof below that suggests to whom one could attribute this metric.

The triangle inequality of the alleged metric follows immediately from the following theorem (simplified version) of Gel'fand and Naimark [2] (see discussion in [1,3] too), so I would essentially attribute it to them. I don't know to whom can one attribute the simple results from majorization theory that are needed while applying the theorem below.

Theorem (GN [2]). Let $A$ and $B$ be complex matrices. Then their singular values satisfy the majorization \begin{equation*} \log \sigma(AB) - \log \sigma(B) \prec \log\sigma(A), \end{equation*} where $s(X)$ denotes the vector of singular values of an operator $X$.

Let $d(A,B)$ be defined as in the OP. Then, we immediately have the following result.

Theorem (Metric). Let $A$, $B$, and $C$ be arbitrary complex matrices. Then, \begin{equation*} d(A,B) \le d(A,C) + d(B,C). \end{equation*}

Proof. From Corollary GN it follows that \begin{equation*} \log\sigma(A^{-1}CC^{-1}B) \prec \log\sigma(A^{-1}C) + \log\sigma(C^{-1}B). \end{equation*} Since for $x \prec y$ it follows that $|x|\prec_w |y|$, from which it then follows that \begin{equation*} |\log\sigma(A^{-1}CC^{-1}B)| \prec_w |\log\sigma(A^{-1}C)| + |\log\sigma(C^{-1}B)|. \end{equation*} It is easy to show that (e.g., Example II.3.13 in [1]) if $\Phi$ is any symmetric-gauge function, then, whenever $|x| \prec_w |y|$, we have $\Phi(x) \le \Phi(y)$. Selecting $\Phi(x) = \|x\|_p$, and applying to the above majorization we immediately obtain the desired triangle inequality.

REFERENCES

[1]. Matrix Analysis, R.Bhatia

[2]. The relation between the unitary representations of the complex unimodular group and its unitary subgroup, Izv. Akad. Nauk SSSR Ser. Mat. 14(1950), 239-260

[3]. The eigen- and singular values of the sum and product of linear operators, A. Markus, Uspekhi Mat. Nauk, 19:4(118) (1964), 93–123.