Many a day/man

What is the difference between

many men/ many a man?

I have come across this explanation in another forum. Does it hold water to you?

"Many a ____" is generally used when you are referring to something that happens often, but not at the same time or in the same place.

Many men were seasick during that voyage. (All on the same trip.)
Many a man has gotten seasick on that boat. (On many different trips.)

Many vases were broken during the earthquake. (in my warehouse)
Many a vase was broken during the earthquake. (all over town)