That was a standard kid’s song when I was growing up (1960’s, Seattle area). I also remember the second line as “That’s my name, too” . We finished off with a line of “Tra la la la la” or “Dum dee dum dee dum”
We just repeated the same words over and over again. Besides being fun to sing, it had the extra benefit of driving our parents crazy.
The earliest reference to it that I know of is, it was sung on an episode of The Andy Griffith Show when it was still in black and white, so it would have been around 1965. It was actually sung slightly differently from how I learned it; normally, while each verse gets softer and softer except for the part between the verses, in this one, most of the second verse is quiet, but when they get to “the people always shout,” they shouted the name.
The only other time I saw it on TV was an episode of One Day at a Time, where it was done the usual way.
I could have written this post but I would have to amend it to read “I grew up in upstate NY, Connecticut, New Jersey and Florida in the 1970s and early 80s.”
We learned this and other camp sort of songs, such as “I Love the Mountains” and “Rise and Shine and Give God the Glory Glory” at PS 24 in the Bronx, early 70’s.