The chance to appear having exactly the same size is zero because the sizes the moon can have are infinite. But it can quite well have almost the same size.
The sun and the moon would look and behave so different so I see no reason for the confusion.

ASTRONOMICAL!!

To turn the question around, the fact that from our perspective sun and moon are practically the same size is a humungous coincidence. A miracle, if you like. An earth eclipse of sun or moon is therefore a sight to behold even on a universal scale.

I don't think anyone could answer that question!
In our own solar system the sun and the moon look the same size from Earth's surface but only at this particular moment in time.
Our moon was allegedly closer to the Earth once so would have looked larger than the sun from here, but it's moving away so in so many millions (billions?) of years it will look smaller than the Sun.
So even if there were another similar system it would probably only stay that way for a certain lenght of time.

Somewhere between 0% and 100% - we don't have enough of a sample set to give accurate estimates of the probability of finding a (life bearing) planet that has a moon that satisfies those distance:size ratios. It could be quite common, or it could be quite rare. I'd lean towards the "rare" side of things as the other moons in our solar system don't always follow the pattern.

Note that this is only really valid "up close" to the sun, as - beyond a point - it'll appear as a teeny spot, rather than a disc of measurable size (I seem to recall pics from a probe that was knocking around Jupiter that only showed a dot for the Sun)

It's just a happenstance, but it could occur anywhere. If the Earth were further away from the Sun, but the Moon was smaller yet in the same orbit, we would have the same effect. Similarly, if closer to the Sun, but with a larger Moon, we would have the same effect.

And there is an infinite number of conditions, where combinations of the size of the Earth's orbit, the size of the Moon's orbit, and the size of the Moon could lead to the same situation of equal apparent diameters.

Of course, there is an infinite number of conditions where the apparent diameters could be different. So the probablility figure you are seeking would be the ratio of two infinite sets. There are situations where that probability can be computed, but it's too complex to go into here.

(Incidentally, to show what I mean, there is an infinite set of even numbers. There is also an infinite set of integers. What is the probability that any integer, randomly chosen, will be even? Obviously, the probability is 50%, the ratio of those two infinite series. That is analogous to the question you posed.)