Is it just coincidence that the Moon follows the ecliptic?

It is not a coincidence at all! (There was, however, some statistical chances involved.) A key question to answer before answering your question, though, is Why is there an ecliptic at all? It is fair to see all the planets and a majority of the smaller bodies existing in a nice flat plane (the ecliptic) and think that it didn't have to be that way. Why not have different planets orbiting in different planes, or even backwards? To get the answer, we have to go back to the origins of the Solar System, when it was still a collapsing cloud of interstellar gas and dust. As gravity collapsed the cloud, the small amount of angular momentum it had became more significant, much as an ice skater pulling her arms in while spinning will make her spin faster. So, with the help of friction, the early Solar System flattened and had a 'spin' in a preferential direction.

As the Solar System evolved, planets formed, and all of them had ingrained in them this spin. Today, we see that in not only the existence of the ecliptic and the fact that planets orbit in the same direction, but in the fact that most planets rotate in the same direction as the sun does (Uranus and Venus being notable exceptions) and most large moons orbit their planets in the same direction (Triton around Neptune being a notable exception).

What about the moon? A leading theory for its formation is that it formed from the debris that resulted in a collision between a still-forming Earth and another planet-sized object. Statistically, both these objects would be most likely to have similar angular momenta to that of the overall Solar System (though it was by no means guaranteed), so their resulting debris would have it as well. When that debris coalesced into our moon, it too would also keep the original angular momentum 'signature,' and its orbit would thus be comparable to that of the ecliptic.

Of course, this is all a very chaotic and statistical process. While angular momentum is always conserved, there are ways of exchanging it between bodies (via gravitational interactions & collisions). Consequently, there are variations away from the ecliptic: the moon is inclined 5 degrees from the ecliptic; Earth's rotation is inclined 23 degrees from it; smaller bodies like comets will sometimes orbit well outside the ecliptic; etc. But the general trend holds, and is a marker you can see present throughout the solar system.

As an interesting aside, while all major planets in our Solar System lie roughly along the ecliptic, that is not always the case for other planets in other star systems. Wasp 17-b, for example, is observed to have a retrograde orbit around its host star, while others have been observed to orbit outside of their own system's ecliptics entirely.