It's commonly understood that when the moon sets and rises again the next day, that's not a "new moon" that is the same moon coming around in the sky again. But the moon does a cycle of waxing and waning again every month (well, 28 days), and then you get a "new moon".

So "one moon" = one cycle of moon phases = roughly 1 month. This is where months came from.

Nope, a "moon" was a single cycle of the moon through its phases, which is closest to a month out of the units we use currently.

While you can ignore that and use the word however you want, and it's definitely possible that people have done so as a form of word play to indicate shorter units of time, it does have a usage that's been around for a least a couple hundred years in English, and way longer in other languages.

The word month comes from moon, and in other languages, the words for month are usually also derived from their words for moon.

In English, the way the word evolved, a it was the period of time from one "new" moon to the next.

Many moons, as a phrase, came from a native American term that was used to express "a long, but undetermined time ago". It isn't exclusive to any specific peoples, nor only to native Americans, but the English idiom version came from a translation from a native speaker

Trade is, however, a similar term for "a long time" that's used almost exclusively an an exaggeration, "a month of Sundays". In a literal sense, that would mean approximately 30 Sundays, obviously, which isn't even a full year, but it's almost always used to express a much longer, but unspecified, time frame.

You've never heard of a "new moon?"