Lady Stallions!?! Not only a New Jersey soccer team, but an oxymoron of Shakespearean dimension. Not a simple oxymoron either, like Romeo's "cold fire, sick health," but a figure of speech on the far frontier of Oxymoronia, along with such classics as Utah Jazz and Christian Science. What, pray tell, is a Lady Stallion? Even in an age of gender fluidity, it's hard to get a grasp on this one. Oh, I understand the principle -- the men's team is the Stallions, and therefore it follows as night the day that the distaff team acquires the same name, prefixed with Lady. In the same way, we have, out here, the Buffs and the Lady Buffs. I would offer not the slightest demurral if the men's team was called the New Jersey Horses and women's team was the New Jersey Lady Horses. It's awkward but logical. Lady Horses may not be either a poetical or an intimidating moniker, but it doesn't boggle the grammatical mind. But a stallion is archetypically male, the snorting captain of his harem-herd of equine odalisques, groupies, wives, mistresses, and girlfriends. Against all challenges, he bites and kicks and if necessary injects himself with steroids and horse growth hormone in order to maintain his macho primacy. A lady stallion is therefore utterly impossible both emotionally and linguistically. The term is a travesty. Better to call them the New Jersey Mares, which makes linguistic sense. Or even better, the New Jersey Geldings (which are, in an odd kind of a way, no-longer-male stallions).
Although I myself would reserve the name Geldings for institutions named after gentleman who have made the sacrifice and earned the right: the Origen Divinity School Geldings or the St. Abelard Preparatory School Geldings. But that's a horse of another color.
Although particularly egregious, Lady Stallions do not stand solitary in their linguistic purgatory. Also competing are the Lady Rams (suggested alternative: Sheep), the Lady Bucks, and the Lady Bulls (alternative: Cows). No teams named the Goats. No Lady Boars (or Pigs either, for that matter). In defiance of all logic, there are the Oregon Lady Ducks, which is truly an impossibility, because properly, the men's team should be the Drakes, the women's the Ducks (or Lady Drakes). There are no teams, as far as I'm aware, named either the Geese or the Lady Ganders; yes, there are the South Carolina Gamecocks, but I'm going to guess that the distaff side is not named the Lady Gamecocks. Wait, I'll google it. OK, the South Carolina women's softball team is named ... the Gamecocks, which is bad enough, but could have been worse, because in my world of propriety, "lady" and "cocks" should never inhabit the same sentence.
Dogs are a problem. No one calls their team the Dogs. But there are some breeds who are allowed to play. Bulldogs and Mastiffs and Greyhounds are in, Poodles are out. The University of Georgia proffers the Lady Bulldogs, but not one college, high school, or club has thought to give us either the Lady Dogs, or whatever is that word by which female dogs are denominated.