I think the latest an embryo can split into twins is in the late blastocyst stage, around 8 days after fertilization. Unlikely that Sharon’s mom found out about the pregnancy before then. But in any case, the positive pregnancy test wouldn’t indicate a particular one of the twins. The chemicals it checks for are hormones present in the mother’s body during pregnancy, not something produced by a specific embryo.
I guess the better question would be, when did she find out it was twins, and was it on an ultrasound, and, if so, whose picture did she see first on it? (And is there any way to know, at this point? I mean, they look exactly alike, and they had plenty of time to switch positions between then and birth.)