Stud comes from a Germanic root meaning a herd of horses or the place where horses are kept for breeding. The same root is behind the Modern German Stute, ‘mare’, Gestüt, ‘stud farm’, and the name of the city of Stuttgart.
A stud farm or (horse) stud is still where horses are selectively bred. Before artificial insemination, owners of mares wishing to have their animals covered by (‘bred with’, since 1530s) a particular stallion would pay a stud fee and bring the mare to the stallion’s location, which was likely to be a stud farm. All such pairings are recorded in a stud book. The location, stud, and the activity taking place there, breeding, began to merge.
In the early 19th c., the word began to denote a breeding stallion. Thus the term out to stud is gendered for many speakers — and the reason stud was transferred to certain human males by the end of the century.
If your selection of a stallion is not for purchase but rather for use on one of your mares, you will also be interested in the management at the stud at which he stands. There are several things that will concern most owners selecting a stud to send their mare to, and these will be discussed in turn. — Mina Davies Morel, Equine Reproductive Physiology, Breeding and Stud Management, 2015.
Stallions available for breeding are said to be standing at stud or at stud service. It makes economic and practical sense for the mares to come to the stallion, who in a given season can cover a number of mares, and that horsebreeders would offer their best breeding stallions on a stud farm.
The financial outlay can be considerable. In 2017, for instance, it was reported that a thoroughbred stallion covered up to 125 mares a year for a stud fee of $300,000 each.
A broodmare is also called a stud mare, and a mare as well as a stallion can be retired to stud after a racing career:
Chesnut Annie, one of the most successful point-to-pointers of the last decade, has been retired to stud at the age of 14. The tough, gutsy mare … brought in the crowds wherever she ran. She was the winner of 38 of her 58 races between the flags and notched up a further 14 placings. She also won two hunter chases.
Annie will be covered by the successful thoroughbred sire, Dr Massini, who is responsible for many top performers including the high class Grade 2 winner Rocky Creek. — Horse and Hound, 15 May 2015.
This almost reads like a wedding announcement.
Out to stud, however, is almost always associated with male animals, though a few horsebreeders do use the term for mares:
If you're planning on sending your mare out to stud, or to an AI centre, you'll obviously want to make sure that the place is run well. — Horse and Hound Forum post, 31 July 2007.
They don't want to have a lag time for the mare to raise her foal, so they put a lesser mare out to stud at the same time and use her for milk. — Dellani’s Choice (blog), 11 Dec. 2011.
One horse enthusiast finds this usage objectionable:
At this point I'm going with Eurodressage because apprently horse and hounds (sic) think you put a mare out to stud. — Chronicle of the Horse, forum post, 25 Jan. 2010.
The vastly more frequent expression for mares is simply to breed:
Perhaps the ideal age to breed a mare for the first time is when she is 3 to foal at 4, or 4 to foal at 5. However, there are many factors that can change that ideal age. — Equisearch.com, 11 Aug. 2011.