Thanks, Mo.

I remember playing "A My Name Is" in Atlantic City, New Jersey, in the 1950s.

It was chanted as you described though I'm betting that the standardized names and objects that kids used for the alphabet differed around the world.

For instance, I remember singing:
A my name is Alice
My husband's name is Al
We live in Atlantic City
Where they sell apples/

{My name wasn't Azizi back then. If it had been, I would have used that name}.

Also, we sang this when we played individual ball bouncing {bounce ball on the ground,and catch it. A variant of this was to bounce the ball underneath one leg and then catch it and then alternate by bouncing the ball under the other leg}.

We never sang this when we played catch with other people.