When it was first invented, Hargreaves spinning machine was know - as were most working implements - as an "engine". It did not take long for the "Spinning Engine" to become, the "Spinning Jenny", in much the same way that the Cotton Engine became the Cotton Gin.
The name was well in place before the spinning mule was invented - and so far as I know, the mule was never referred to as a "jenny", presumably to avoid confusion between the two.

It was called a "mule", incidentally, because it's technology was a "cross-breed" between two different sort of machine.