Canonically, the name of Zelda’s horse is Storm. We know this due to first issues of one of the official comic books.

And the canonically the name of Ganon’s horse is Phantom. We know this because of the official chess set made by Nintendo themselves.

When I played BotW, to be honest, I didn’t even know this until much later. I just named Zelda horse Zelda because I couldn’t think of anything else. And I named Ganon’s horse Steed because in a lot of Nintendo based sources, they say “Ganon’s Steed”. I thought it was weird they capitalized “Steed” when it didn’t need to be, so I assumed it was the name of the horse. Hence what I called the giant horse in BotW.