I noticed that even if I have an approved booking for Mare A for say, March 1st, if the mare is at the same farm as the stallion, I can breed her now. Is this supposed to occur?
Yes. =) It won’t happen until said date, but this is a concession made for the game because it IS a game. Otherwise, you would have to log onto the game on the EXACT breeding date and click “breed this mare” then…which would be a pain, especially for those bigger stables!
Ah, okay, that makes sense 
So it’s safe to breed a March booking and not risk the foal arriving in December due to the current RL date?
Yep. The foal will be bred in March and should arrive in February sometime.
Basically “Breed this Mare” is equivalent to “setup everything in order for the breeding to happen”, and then the game acts as if everything did on the breeding date. 
The one thing you can’t do is use a booking that’s already past. So in June you can’t use a May booking.