Before moving on to the question, let me clarify what I mean by the terms "total translational kinetic energy" and "average kinetic energy of a molecule". The total translational kinetic energy $K$ of all the molecules of the gas is
$$K=\sum\frac 1 2 mv^2=\frac 1 2 mN\frac{\sum v^2}{N}=\frac 1 2 Mv^2_\mathrm{rms}$$
where $m$ is the mass of a single molecule, $v$ is its velocity, $N$ is the total number of molecules in the container, $M$ is the total mass of the gas sample, and $v_\mathrm{rms}$ is the root-mean-square (RMS) speed.
The average kinetic energy of a molecule is
$$K/N=\frac 1 2 \frac M N v^2_\mathrm{rms}=\frac 1 2 mv^2_\mathrm{rms}$$
where the symbols have the same meaning as of the previous case.
It is said that, at a particular temperature, different gases have the same average kinetic energy.
Mathematically,
$$\frac 1 2 m_1v_1^2=\frac 1 2 m_2v_2^2$$
where $m_1$, $m_2$ are the masses and $v_1$, $v_2$ are the rms speeds of the two gases respectively.
As we discussed at the beginning of the question the average kinetic energy of a single molecule is much different from the total translational kinetic energy of all the molecules of the gas. I don't understand why the average kinetic energy of a single molecule must be the same for all gases at the same temperature instead of the total kinetic energy of all the molecules.
I understood that the absolute temperature $T$ of a given gas is proportional to the square of the RMS speed $v_\mathrm{rms}$ of its molecules as per the following equation:
$$T=\left(\frac{273.16~\mathrm K}{v^2_\mathrm{tr}}\right)v^2_\mathrm{rms}$$
where $v^2_\mathrm{tr}$ is the RMS speed of the molecules at $273.16~\mathrm K$ (triple point of water). Further, both the total translational kinetic energy and the average kinetic energy of a molecule are proportional to the square of the RMS speed. Due to this similarity, I'm unable to see which factor is responsible for the same average kinetic energy instead of the same total kinetic energy at the same temperature for two different gases.
So, why do different gases have the same average kinetic energy at the same temperature instead of the same total kinetic energy?
While searching this site for this doubt, I came across these questions - Temperature and kinetic energy of molecules, Does the same temperature imply the same translational kinetic energy?, If two gasses are in thermal equilbrium, do their molecules have same amount of Kinetic energy?, and Intuitive explanation why rate of energy transfer depends on difference in energy between two materials?. But I didn't find any distinction between the two types of kinetic energy discussed in the question and hence the reason for the choice of average kinetic energy over the total translational kinetic energy.
Initial part of the question is based on the chapter "Kinetic Theory of Gases" from the book "Concepts of Physics" by Dr. H.C.Verma.