Relation of translational and rotational diffusion of a spherical particle

Yes, your interpretation is correct. This surprising coincidence can be understood with the fluctuation-dissipation theorem, which describes the Brownian motion for any degree of freedom in a thermodynamic system.

The theorem relates Brownian motion for a coordinate $x$ to the energy dissipation caused by changes in $x$. In particular, if two coordinates $x_1$ and $x_2$ cause the same energy dissipation when varied, then they will have the same diffusion coefficient.

How does this relate to our problem? For a rotating sphere, we have six degrees of freedom - three of them translational and three rotational. When the sphere is translated or rotated, there will be a flow induced in the surrounding fluid, causing energy to be dissipated through viscosity. This is precisely the "energy dissipation" that the theorem deals with!

Let's compare a translational degree of freedom $x$ to a rotational degree of freedom $\theta$ in terms of their energy dissipation. If we vary $x$, the fluid will flow with the Stokes flow past a sphere (link), dissipating energy at a rate $P_{x} = 6 \pi \mu r \dot{x}^2$. If we vary $\theta$, the fluid will flow azimuthally, dissipating energy at a rate $P_{\theta} = 8 \pi \mu r^3 \dot{\theta}^2$ (derivation, page 6). In other words, moving through the liquid at speed $v = \dot{x}$ dissipates $\frac{3}{4}$ as much energy as rotating the surface at speed $v = r \dot{\theta}$.

With this fluid dynamics factoid, we can go back to the fluctuation-dissipation theorem to see the diffusivity in $x$ must be $\frac{4}{3}$ of the diffusivity in $r \theta$. The origin of this result is now clear - it basically comes down to the similarity between two fluid flows.