Does the Honey moon have a precession problem? In Steven Colbert's interview with Neil deGrasse Tyson after about 03:16 the following is explained:

But wait, all the moons - all the full moons of the year have names. Some are familliar like the Harvest moon… and the Moon in June never gets too high in the sky, so it takes on sort-of an amber sunset color, the whole trip. And so that's called the Honey moon. So the June full moon is the Honey moon.

Is there a problem here though? For any given month won't the full Moon oscillate between low and high above the horizon at midnight on an 18.6 year cycle?
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Is there a problem here though? For any given month won't the full
Moon oscillate between low and high above the horizon at midnight on
an 18.6 year cycle?

No. The full moon is always opposite to the sun (within 5.2 degrees, the angle between the moon's orbit and the ecliptic). So, when in summer the sun is high in the sky, the full moon will be low, similar to the height of the winter sun.
You are correct about the oscillation with the 18.6 year cycle, but that oscillation amounts to only +/-5.2 degrees, while the summer/winter difference results in +/-23.5 degrees. So (in northern latitudes) the typical summer full moon will be 47 degrees lower than the typical winter one, and even the highest possible summer full moon will still be 37 degrees lower than the lowest winter full moon.