Azizi, the song goes back a long ways, known in many European countries, according to Randolph in Ozark Folk Songs. You are interested in children's singing games, so here is a children's version from Georgia, Newell (1883, 1903), Dover reprint.

Lyr. Add: MY HOUSEHOLD

I had a little rooster, and my rooster pleased me,
I fed my rooster beneath that tree;
My rooster went- Cookery-cooery!
Other folks feed their rooster, I feed my rooster too.

I had a little lamb, and my lamb pleased me,
I fed my lamb beneath that tree;
My lamb went - Ma-a-a!
Other folks feed their lamb, I feed my lamb too.

and so on with other beasts.
"The names of animals being distributed among children, one, in the center of the ring, sings the words; at the proper point the child who represents the animal must imitate its cry; and as at each verse the animals who have already figured join in, the game becomes rather noisy."
Newell cites a German version, "When I was a poor woman, I went over the Rhine: my goose was called 'Wag-tail', my maid said 'So he said', my pig 'Lard-pot', my flea 'Hop-i'straw'," etc.

From W. W. Newell, "Games and Songs of American Children," no. 48.