Just my observation as an outsider but I think there is no direct equivalent because historically women have not been groomed for power in the same way as men, so girls' schools didn't gain reputations for the same kind of things as boys schools.
I had a few very wealthy acquaintances at University. The boys had been to Rugby and Harrow (don't think I knew any old Etonians) and the girls to Cheltenham Ladies' College and Roedean. They were from very similar backgrounds and definitely did see themselves as the ruling class, but I got the impression that the girls, though they had great academic credentials, had not been brought up/educated to see themselves as future leaders, but rather as potential wives of future leaders. The one girl from that kind of background that I knew who was really focused on her own career progression had been to Rugby for 6th form in fact.
Obviously this is a very small sample, and it was the 80s so hopefully things have changed, but I suspect the reason why there is no true "girls' Eton" is good old fashioned misogyny.
(I hasten to add that I don't subscribe to the idea that all our leaders should come from such schools, but that's a different issue altogether. )
