Is $\pi^2 \approx g$ a coincidence?

The differential equation for a pendulum is

$$\ddot{\phi}(t) = -\frac{g}{l}\cdot\sin{\phi(t)}$$

If you solve this, you will get $$\omega = \sqrt{\frac{g}{l}}$$ or $$T_{1/2}=\pi\sqrt{\frac{l}{g}}$$ $$g=\pi^2\frac{l}{T_{1/2}^2}$$

If you define one metre as the length of a pendulum with $T_{1/2}=1\,\mathrm{s}$ this will lead you inevitably to $g=\pi^2$.

This was actually proposed, but the French Academy of Sciences chose to define one metre as one ten-millionth of the length of a quadrant along the Earth's meridian. See Wikipedia’s article about the metre. That these two values are so close to each other is pure coincidence. (Well, if you don't take into account that the French Academy of Sciences could have chosen any fraction of the quadrant and probably took one matching the one second pendulum.)

Besides that, $\pi$ has the same value in every unit system, because it is just the ratio between a circle’s diameter and its circumference, while $g$ depends on the chosen units for length and time.

It's annoyingly unclear how far it's a coincidence, but at any rate it isn't completely a coincidence.

As you can see in e.g. the Wikipedia article about the metre, a unit almost equal to the metre but derived from a pendulum was first proposed in 1670 and the idea was in the air when revolutionary France decided to make a new set of units.

This pendulum-derived unit, if adopted, would have made $g$ equal to $\pi^2\, \mathrm{m}/\mathrm{s}^2$ by definition. The proof of this is very easy, and can be found in other answers here, so I shan't repeat it.

(So if that definition had been adopted, the answer to the question here would be an unequivocal yes.)

Here is a link to an English translation of the report of the commission appointed by the French Academy of Sciences. They explain that the pendulum-based definition is very nice but has the drawback that it depends on the second which is a rather arbitrary unit. So instead they propose to take $10^{-7}$ of a quarter of a meridian of the earth.

Now, this unit they've adopted is (1) almost exactly equal to the pendulum-based unit, but also (2) derived in a conspicuously simple way from the dimensions of our planet. So it's overdetermined. Did Borda, Lagrange, Laplace, Mongé and Condorcet (an impressive list of names indeed, by the way!) choose the particular earth-based definition they did because of its closeness to the pendulum-based definition, or not? That's what's annoyingly unclear.

I find two useful clues in their report. They point in opposite directions.

First, they say

in adopting the unit of measure which we have proposed, a general system may be formed, in which all the divisions may follow the arithmetical Scale, and no part of it embarras our habitual usages: we shall only say at present that this ten millionth part of a quadrant of the meridian which will constitute our common unit of measure will not differ from the simple pendulum but about a hundred and forty fifth part; and that thus the one and the other unit leads to systems of measure absolutely similar in their consequences.

which makes it plain that they knew how close the two were, and were glad to take advantage of it. But, second, they say

We might, indeed, avoid this latter inconvenience by taking for unit the hypothetical pendulum which should make but a single vibration in a day, a length which divided into ten thousand millions of parts would give an unit for common measure, of about twenty seven inches; and this unit would correspond with a pendulum which should make one hundred thousand vibrations in a day: but still the inconvenience would remain of admitting a heterogeneous element, and of employing time to determine an unit of length, or which is the same in this case, the intensity of the force of gravity at the surface of the earth.

so they were clearly prepared to countenance the possibility of a somewhat different unit, and the argument they give against this one has nothing to do with its disagreement with the pendulum-based unit that's so close to the metre.

(Though ... if they had ended up plumping for that definition, they might also have proposed redefining the second to be $10^{-5}$ days, in which case they would again have made $g=\pi^2$ by definition.)

I think the first of those passages is enough to make it clear that it isn't a total coincidence that $g\simeq\pi^2$. Borda et al knew that their definition was close to the pendulum-based one, and offered that fact as a (minor) reason for accepting it. But I think the second is enough to suggest that it could easily have been otherwise: my feeling is that if the definition based on meridian length had been, say, 5% different from the pendulum-based one, they would still have preferred it.

In comments below, user Pulsar has found an interesting article about this whose conclusions are roughly the same as mine: it sure looks as if the pendulum-based second was a motivation for the choice of $1/(4\times10^{7})$ of a meridional great circle, but nothing here is altogether clear and we have to rely on conjecture about the motivations of the scientists involved.

In addition to Anedar's answer, I'll try to address things from a bigger perspective.

When they made the SI unit system, they chose units that are convenient for humans. From a scientific perspective, it would make sense to use for example Gaussian units, but for the man on the street, it is more useful to have the units more or less agree to things humans encounter in daily life.

So for a length scale, you want something that is roughly the same size as a human body. And for a time scale, you want something that roughly represents how fast humans can count.

In the SI unit, the meter and second were chosen in such a way that:

• 1 m is roughly the typical distance of an adult's leg.
• 1 s is roughly the typical time it takes to walk two steps.

Because there is a relation between leg length, time between steps and gravity, this corresponds to a gravity constant in this unit system of roughly $\pi^2$.

They could have chosen different anthropocentric units. The meter could have been defined twice as long or twice as short, and the second could have been defined twice as long or twice as short. But not a factor thousand longer or shorter, then it would not have been accepted by the community, because the units would have been very inconvenient.

So I claim that on every planet with life which developed a system of units, their local gravity constant in their system of units is between 1 and 100.

(This answer explains why the numeric value of $g$ in SI units is not 10000000. It does not explain why the numeric value of $g$ in SI units is not 13.)