I have never seen any reference that suggested that Rangoon oil ever contained sperm oil. Sperm oil was used as a lubricant for guns, watches, and other fine mechanisms because it had no tendency to gum up. It was also used by gunsmiths for tempering springs. The old German gunsmith who taught me spring tempering had a good supply of it with a picture of a whale on the label of the glass jugs.
KY Jon's initial post provided the best answer for what Rangoon oil was. It seems unlikely that the source today would be the same natural oil seep near Rangoon, Burma that was used in the 1800's, or that the same early refining technique would still be used today, even though fractional distillation is still used. The original Burmese petroleum base stock seems to be an important part of the recipe. Holland & Holland Rangoon oil is advertised as containing synthetic esters, so we can be sure their formula is not the same as the original product.