Mathematical coincidence of the Schwarzschild radius of the Universe?

I read here that a black hole with a mass of the observable universe, $M=8.8times10^{52}kg$, would have a Schwarzschild radius of $r_s=13.7$ billion lightyears.

I immediately noticed that at the speed of light, to travel a distance of $r_s$, it would take nearly exactly the age of the universe. Is that a coincidence, or is there a connection?

Normally, I’d brush this off as a coincidence and go on with my day. However, the top answer to this post details that the previous values are related to the Hubble constant $H_0$; the reciprocal of which is known as Hubble time, only varying in value from the age of the universe (due to the non-linear expansion of the universe), by a dimensionless factor of about $0.96$

Thus, considering that $r_s$ is related to $H_0$, which is related to the age of the universe, is that value of $13.7$ billion (light)years a coincidence, or is there a direct mathematical relationship?

As an undergraduate in math, I’m not extremely familiar with these concepts, and may have glossed over something obvious to the more atrophysically atuned. Hence, this is mere curiosity.