Given our knowledge of physics, it must surely be a coincidence.
$\pi$ arises from investigation of certain mathematical relations. Let's consider the ratio of a circle's radius and circumference: This is not the most interesting occurrence of $\pi$ (the relations studied by Euler and after are sometimes considered more noteworthy) but it is a nice simple example.
The circle is a set of points equidistant from an origin. The constraints only allow a single arrangement of such points in certain kinds of space. $\pi$ characterizes this unique arrangement. Note how this was arrived at with purely mathematical reasoning, without any recourse whatsoever to the physical world. Aliens in an entirely different parallel universe, or demons in Hell could have reasoned likewise, and discovered the same $\pi$, regardless of how different the physical laws governing them are. Mathematics is unaware of reality, it doesn't care about what happens in the so-called real world. All it is, is logical deduction of consequences arising from a set of axioms.
It so happens that $\pi$ can be experimentally observed, for instance by constructing circles out of wire. But there is a very biased causality here: The wire loop exhibits $\pi$ because our world is like the Platonic ideal of Euclidean sapce, not the other way around. Although it is worth noting that there is of course a reason why Euclid happened to start with exactly that kind of space, and not another.
$g$ arises from the action of masses on each other. For unclear reasons, the world which we inhabit contains masses. The way these masses behave appears to follow certain rules. These rules were deduced by observation of the physical world. Analysis of the rules yielded $g$.
Aliens in Universe X, or demons in Hell, could not themselves find $g$ without observing our universe. By methods analogous to ours, they can find $g_{alien}$ or $g_{hell}$. These can easily be different from our $g_{earth}$, but they're not prohibited from coincidentally equaling it either. The numbers have absolutely nothing to do with each other, being parts of disjoint physical systems.
Note that physics is itself an abstract construct, there is no reason to believe that the universe obeys our laws of physics, it has merely never been observed to act in contradiction to those laws which have not yet been refuted (the circularity is meaningful). Unlike mathematics, the construct of physics relies on not only a priori assumptions, but also a posteriori observations of our physical world.
You don't have to accept that $\pi_{earth}=\pi_{hell}=\pi_{alien}$. You can, for instance, make the disturbing but reasonable objection that mathematics is nothing but an artifact of the human brain, and is not universal but a posteriori. In a sense this position weak, because we have observed animals to have a similar understanding of mathematics, but of course that is only circumstantial evidence, not proof.
If you do accept that $\pi_{earth}=\pi_{hell}=\pi_{alien}$, being that $\pi$ is obtained with no input from the physical world: Then whereas everyone's $\pi$ is necessarily equal, everyone's $g$ need not be. Thus the relation you observe would only hold in our universe, not in hell or Universe X. In other words, our universe "easily could have had" a different $g$ - it is unclear whether the laws of physics we know had no choice but to be the way they are, or if there was some kind of dice rolling to conjure up a bunch of random laws, and we "could have" ended up with a different set. It is not even clear if the laws have always held, or will hold in the future. Although we have never observed them not holding so far - except for the ones we did, but we don't talk about those anymore...
One can observe that while mathematics doesn't care about the world, the world does appear to obey mathematics. We have never observed the real world contradict mathematical logic. So, it is not impossible that one day, the true nature of $g$ will be understood, and it will turn out to have a geometrical origin (for instance), and we will find out that your observation is in fact meaningful, not mere coincidence. But as far as I know, no such geometrical explanation exists.
Note 1: In this answer, I took a philosophical position regarding the nature of mathematics, that is not understood to be necessarily true. There are valid objections to it. I personally feel that my position is prima facie congruent, so I wrote this answer. If you have a radically different concept of mathematics, perhaps other answers can be given to your original question.
Note 2: I didn't want to be mean and give you the boring answer right off the bat. For the sake of completeness, here it is: $\pi^2$ is like $g$... only if you use meters and seconds, two explicitly arbitrary units. In Planck units the relation does not exist. In fact, with the right units, you can make $g$ be like $e$, or your age, or your ZIP code, or any other number you desire.