Do Ganondorf's Steed, and Zelda's Steed have names?

Directly stated in the games, no. The horses don't have any names.

However, outside of the games, they kind of have names. There's a white steed named Storm that Zelda rides in the Valiant Comics that's very similar to the in-game horse that Zelda and Impa ride out of Hyrule Castle, although the comics are mostly independent works with no relationship to the games. From the wiki:

Storm is the name of a large, white horse which belongs to Princess Zelda, as seen in the Valiant Comics. Storm helps Zelda leave Hyrule's North Palace when she attempts to leave Hyrule in order to keep the Triforce of Wisdom from the reach of Ganon.

Ganondorf's horse name is a little trickier. I've seen numerous unproven citations that the horse's name is Galloughs ranging from GameFAQs posts to Reddit posts to DeviantArt fanfiction, feel free to google the name yourself, and the most consistent proof I can find is that multiple people seem to think the Ganondorf horse object in OoT's code is named Galloughs. However, looking at multiple object lists here and here, the horse is just referred to as Ganon's Horse or something equally uninteresting, and the dataminers of The Cutting Room Floor haven't made mention of any horse-related discoveries, so I'm personally skeptical.

Zelda's horse name is Storm which can be seen in the comics. Ganon's horse name is Phantom which is also in the comics.