I think some slightly similar sounding terms got mixed together, and I don't think they're related. I can't prove it, and I may very well be wrong. I heard the 'Puddin' Tane' rhyme when I was a kid, too. Poontang isn't a word that fits in a kids' rhyme.

Poontang was a word I heard from the Vietnam vets I worked with when I first started in the Air Force. I had to get past the, "Oh, no - I can't tell you. It's too rude" stage before they'd talk.

It's pretty funny how far a vulgar term can travel when it's so unfamiliar people don't know it's vulgar. People get fooled into thinking it's a 'nice' word and it squeaks right by censors.

From the Urban Dictionary: Poontang
(just the first entry):

Function: noun
Inflected Forms: plural - poontang; abbreviated - poon
Alternate Spellings: puntang
Etymology: Filipino (Ilokano dialect) putang (whore, fuck)
1: female genitalia
2: sexual intercourse

1: Her poontang was the warmest and silkiest I had ever fucked.
2: I need to get some poontang while I'm down there in Cancun.