I saw these 2 threads:
However, I don't find the answers addressing my question. Einstein relation connects between temperature, $T$, and time, $t$:
$$D \propto T$$
Where $D$ is the Diffusion coefficient and has a time dimensions inherently: $$\left[ D\right] \propto \left[t^{-1}\right]$$
Is it true to say that fluctuation, temperature and time dilation are equivalent?
As noted in the comments time dilation only depends on velocity (special relativistic) and gravity potential differences (GR). Sure, temperature depends on the random velocities of the particles (well, actually kinetic energies, so it could be also from random but cyclic vibrational velocities).
So, yes, you might observe the consequences of vibration frequencies and linear velocities, and those could be relativistic. However, keep in mind those velocities will tend to be random, as much inward velocities as outwards, as much to the left as to the right. So you will normally see, if anything, a spread of frequencies (from the Doppler shifts), and possibly a very small spread of time constants such as relaxation, other time constants and heat transfer times. Doppler shifts/spreads have been observed for a long time now in the spectrum of materials.
Time dilation effects from heat/temperature are hard to detect, maybe too hard, except for special circumstances like clocks - which are not too different from measuring Doppler spreads in principle though different measurements. For instance there have been similar thermal effect seen in materials used as clocks. In this article it was cooled down and the spread in time accuracies were reduced by 50 fold. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.053002. I didn't search for the history of clock thermal adjustments, but I'd guess that some of it has been done over the years.
