How does one treat Dungeon World's Princess class move: Unicorn?

It seems to me that the Unicorn has exactly the same status as a Ranger's animal companion. There are modifiers to the Princesses moves - a Pastel Unicorn adds +1d4 damage to the Princess, for example, in the same way a companion would add its ferocity.

It's present fictionally; a Unicorn follows the Princess around for all to see, and the the rest of the world should behave accordingly.

Everything else it does also the same as for an Animal Companion; the GM makes a move. If the Princess lets the girls of the village brush the Unicorn's Mane, the GM says what happens. If the Princess asks the Unicorn to fight the Cosmic Hamster, the GM says what happens. And by "says what happens", I mean just that. No dice rolling or trading hitpoints; just say what happens. The decisions should be influenced by the Unicorn's Tags, though; do you leave your Savage Unicorn alone with the village girls? Will it attack the Hamster if not Brave? Will it win if it's Deathly? And so on.

The fact that there's a Unicorn described in the Codex is irrelevant (though confusing). They're not the same thing at all. The Princesses Unicorn has no stats beyond those described in the move.

Before I get to the real answer at hand, let's look at the Ranger, a classic class. It has the move Animal Companion. As we know, the Ranger's animal companion has no stats (as in, HP, armor, moves, damage, etc). It gives mechanical benefits (ferocity, cunning, armor, instinct), but most importantly it exists in fiction. The Ranger has a loyal animal companion. In DW, that's all the rules the move really requires: it just also gives other rules about how the Ranger directly benefits from their companion.

So why doesn't the animal companion have stats (HP, armor, moves, damage, etc)? I think there are three reasons for this:

1. This way it exists purely in fiction. The GM does not deal damage to it, the player does not directly control it, no one needs to worry about the moves it makes.

2. This way it cannot be killed without fiction stepping in. Come on, it's the Ranger's best friend and greatest class move. It's not cool to lose a thing like that just because a random encounter goblin happened rolled max damage or something.

3. It's much simpler to write and play this way. If it had, say, HP equal to your level + 4, and damage based on its form, and armor, and moves, and blah blah blah. Word counts would be met, and the players and GMs wouldn't want to read and memorize the class.

In summary, the animal companion cannot be "overpowered" because it has no stats. It is and does exactly as the GM says, based on the GM's principles, of course. Think dangerous, be a fan of characters, and so on.

So let's clarify a couple things about the unicorn:

Does the Princess just simply have a Unicorn, with monster codex stats,

No stats it seems, and especially not from the GM-only monster codex.

and fights on her behalf

I'm quite certain the fiction says that unicorns are peaceful creatures, but that's just how I imagine unicorns. The way I see it, the unicorn can help the Princess in combat situations, but because it doesn't have moves or damage, it's purely fiction-based. That's not to say that it can't help. You just won't be rolling any damage dice for it.

Or is this treated more like a Ranger's Animal Companion?

Not really? I mean the rules don't say anything about the Ranger. Granted, the rules text for the unicorn could have been a bit lengthier, but the Ranger's animal companion is well-detailed because it is expected to function in a very different role compared to the unicorn. A hunting companion, compared to a tea party guest.

Additionally, how do the tags work in relation to the Princess? Do the Color tag buffs apply directly to the Unicorn, or to the Princess (especially in context for the Luminescent tag, which I don't believe that the Unicorn has the Gentle Touch ability?), or both?

Yeah, I think this is where the confusion came from, and it's definitely not spelled out clearly enough. The way I see it, the tags apply to the princess. Because, you know, the unicorn can't add +d4 to its undefined damage, and doesn't have the move Gentle Touch. Note that if the princess didn't have a move called Gentle Touch, I might have understood it as a case of descriptive rules text, as in: "the unicorn, with a gentle touch, heals d6 damage". But that's not the case.