Is this a coincidence?

Actually, I favor the idea that it is not a coincidence, at least it's not a coincidence that those two ratios bear some strong resemblance. The reason is because at the large scale, as seen already, the largest time is we can detect is the age of the universe T, and the largest distances L we can measure are going to be directly related to that age by L~cT, at least in rough order (the expansion of the universe makes distance L a rather nonunique issue). On the smallest scales, the energies E we require to probe the shortest times t obey Et ~ h by the uncertainty relation, but if we assume those energies will be manifested in relativistic particles, the best we can localize those particles in a length scale l obeys E/c ~ h/l, also by the uncertainty principle. Eliminating E gives l ~ ct, and so forming the ratio of the largest and smallest scales gives L/l ~ (cT)/(ct) ~ T/t. So whatever that ratio comes out, the two should not be all that much different.

So that those ratios are similar is not nearly as surprising as that they come out so incredibly huge. Why does our universe exhibit such incredible contrasts of times and distances? One might look to the anthropic principle there, in the sense that life requires both long timescales to appear and short timescales to support the required complexity, but then there's still the issue of whether or not one regards anthropic explanations as explanations at all. Also, even if we take anthropic thinking, it seems likely the universe could have produced intelligent life even if it was much smaller, and could have supported such life using much slower processes than those we can detect in our supercolliders. It appears our universe is built with a significant degree of "overkill" in regard to both the volume available for events to happen, and the smallest possible distance scales on which they do happen.

Giving it some thought, I think this is a question worth some examination, so I voted your question up.

The largest distance we can see and the amount of time we can see by looking as far in the past as we can are directly related. We can see $13.8$ billion light years away, though we can infer that those most distant objects are currently 44 billion light years away (a variation of 3.2), and we can also infer that the diameter of the observable universe is twice that, but let's ignore the expansion and just use the farthest distant object we can observe $- 13.8$ billion light years and $13.8$ billion years. Using those numbers doesn't significantly change your question and those two numbers are clearly related.

Translating to seconds and meters, we get $4.35 times 10^{17}$ seconds (let's not call that nearly $10^{18}$, it's less than half).

And translating light years to meters, we get $1.31 times 10^{26}$ meters. About 300 million times the other, which makes sense cause that's how many meters light travels in 1 second.

OK, let's look at small and see if we get the same.

The Large Hadron Collider gives a pretty precise measurement of the smallest distance we can measure, see this question for details. That number is what you said, about $10^{-18}$ meters.

The shortest time we can measure isn't quite as clear cut. This article gives a time of 250 atto seconds ($250 times 10^{-18}$ seconds):

Most recently, the techniques of pulse generation and measurement have been extended to the ultraviolet and soft x-ray region of the spectrum where pulses on the order of 250 attoseconds have been created and used to study the motion of electrons around the nucleus of a neon atom.

But it mentions that quantum models can infer time periods much shorter. The shortest is the estimated average lifespan of the top quark at about 0.4 yoctoseconds or about $4 times 10^{-25}$ seconds (40 times your estimate of the 1 to the power of -26).

Events as short as about $10^{-25}$ second have been indirectly inferred in extremely energetic collisions in the largest particle accelerators. For example, the mean lifetime of the top quark, the most massive elementary particle so far observed, has been inferred to be about 0.4 yoctosecond.

So, when I run the numbers, I get the shortest estimated time and the age of the universe have a ratio of about $1.1 times 10^{42}$ and the shortest measured distance and the longest observable distance have a ratio of about $1.3 times 10^{44}$. A variation of more than 100. That's pretty close, but nothing remarkable.

Also, I can't for the life of me see what the lifespan of the top quark and the energy they can generate per collision in the LHC would have to do with each other, so I would say that there is no relation at all, and at risk of taking on your author, I would disagree with his claim.

It's indeed a coincidence. The human limits on observables cannot have some deeper explanation.

In fact, we can claim many things like this, without having some sort of deep meaning.

The other thing is they are not "exactly" equal. Their ratios just happens to be in the same order. There are also uncertainties in all these measurements and the ratios are never equal to each other. If they were exactly equal in some sense then we may thought there is some deeper meaning.