In astrology, a bunch of trivia about your life factor into your reading... So like basic newspaper horoscopes are just like "which range of dates does your birthday fall into? Oh okay you're a Pisces then, so go read the horoscope for all Pisces..." But if you go to a "real" astrologer for a "real" reading, they wanna know other stuff, including what time you were born, and that'll help personalize the reading.

Mom is making the assumption that the reason for the question is to use her response as input to an astrology reading. Instead of sharing the time of birth, she warns the subject to stay away from "her," the astrologer, who are stereotypically female.