A friend observed this and I recently confirmed it is fairly accurate (at least for Seattle WA) with the recent full moon set on a clear night now near the winter solstice. The full moon set at dawn near where I have the seen the sun set at the summer solstice, over the Olympic Mountains NW of Seattle. There probably is a connection between the plane of the earth's orbit and plane of the moon's orbit; either a fixed angle or perhaps precession of some type?
Hi Bob. Thanks for the question. It is another head scratcher. Fortunately I love these.
The sun travels along a path in the sky called the ecliptic. The moon's orbit is inclined by 5.1 degrees to the ecliptic. When the moon is full and it sets, it could anywhere from +5.1 to -5.1 degrees off the ecliptic. This varies over time. There are about 12.37 moon cycles in an earth year, so one year from now, the moon would not set in the same place on the same specific day since its inclination will be different than it was the year before whereas the sun will, even though the two solstices are six months apart, since its path *is* the ecliptic. The point here is that at any time of year, the sun is traveling on the ecliptic, while the moon is not. That is the first reason I thought of for there being no connection between the two events in your question.
Intuitively, I know there is no connection. However, if there were exactly 12.000 months in a year, I would be "inclined" to think it was possible or even true. But that is not the case. I do believe there are other reasons why the relationship between these two events was a coincidence and not a rule, but I can't think of any right now. Nor do I think it is the only factor involved in this scenario. I will update the answer if something else comes to mind.
So there you are. My shoot from the hip answer to this question. I hope this helps. In the meantime, here is a site I am going to be looking over. Maybe you should too.