The correct term for this type of dancing as "flamenco." There is no such thing as "flamingo dancing" so when you hear someone saying that, they are merely incorrectly pronouncing "flamenco dancing."
A "Flamingo" is a pink colored bird with long legs, not a type of dance. It is a very common mistake to refer to flamenco dancing as flamingo dancing.
A likely culprit of this is the fact that the Spanish word for flamingo is "flamenco." This likely causes a lot of confusion, especially for those that are familiar with the Spanish language.