This originates from a comic song “Little Bingo” performed in London in c.1780, which started:
The farmer’s dog leapt o’er the stile His name it was little Bingo
In the 19th century it was still being sung by adults with an accompanying dance. By this time, the song had been taken up by children as a singing game, and the dog’s name sometimes appeared as Stingo or Pingo. The singing game is rare now but it is still remembered as a nursery rhyme.