Although they are of different dimensions, the value of the Rydberg energy is very close to that of the Hubble constant.
Rydberg energy (R): $2.179 \times 10^{-18}$ [Joule] = 13.6 [eV]
Hubble constant (H): $2.189 \times 10^{-18}$ [1/sec] = 67.66 [(km/s)/(Mpc)]
Is this just a coincidence?
Or is there an interpretation explaining the closeness of these two values?
It's a coincidence.
To see why this is so, consider this method to get your height to be equal to the Rydberg energy, even though it's of different dimensions. First, define a new unit, new_unit $\equiv$ (your height in meters)/$(2.179 \times 10^{-18}) m$. Then when you measure your height in the new unit, you'll get $(2.179 \times 10^{-18}) new\_unit$.
Therefore we see that the exact value of a dimensionful number is not especially meaningful, since it will depend on the units you are using. The exact value of dimensionless numbers, on the other hand, are significantly more interesting - which is why people investigate if the fine structure constant varies over time or space.
For every situation where
although they are of different dimensions, the value of $X$ is very close to that of $Y$
applies, it is always a coincidence.
Units are human constructs: they are based on arbitrary choices of the lengths, time intervals and masses we used to set up the unit system, and there is nothing intrinsic about them. And indeed, as you've shown, if you use other unit systems, the values do not match.
For more details and further reading, see my (and others') answers to Why is it "bad taste" to have a dimensional quantity in the argument of a logarithm or exponential function?, Is it possible to speak about changes in a physical constant which is not dimensionless? and Have I discovered how to calculate the proton's mass using only integers?, and the links therein.
