Mathematical coincidence of the Schwarzschild radius of the Universe?

It's a very good question, connected to the 'flatness problem' of cosmology.

It can be solved by presuming that, as the universe expands, all objects - people, atoms, galaxies etc... expand too.

This leads to an alternative interpretation of redshift. If the size of atoms and Plancks constant were lower in the past, then from $E=hf$, the energy of photons arriving from a distant star would be lower, hence the redshift.

But crucially it explains the coincidence that you have noted, as follows...

With the symmetric expansion above all physical constants and lengths must change in proportion to the number of length dimensions in them e.g. $h=h_0e^{2Ht}$ where H is a constant related to Hubble's constant. This type of expansion keeps everything in proportion.

Each particle has total energy $mc^2-GmM/R$ where $m$ is the particles mass and $M$ and $R$ represent the mass and radius of the universe (small constants are omitted for simplicity).

During the expansion this becomes $(mc^2-GmM/R)e^{2Ht}$ and energy can only be conserved if $mc^2-GmM/R = 0$,

i.e. if $G=frac{Rc^2}M$

There is a fundamental difference with traditional cosmology, with this approach gravity is caused by the expansion and has the value necessary to conserve energy as the expansion occurs.

The alternative approach naturally predicts that the matter density will be measured as 0.25 or 1/3 depending on how it's measured. This seems to be the case.

There is a link to the theory here Here, https://vixra.org/abs/2006.0209

So perhaps the alternative theory answers your question.