There Were Dragons When I Was A Boy

Many of the dragons in the books don't have names. It's more of a human thing, and the dragons who do have names typically have them because of humans.* Toothless is named by Hiccup (the only dragon named by Hiccup III in the books, as far as I'm aware - let me know if I'm mistaken). The Deadly Shadow, like the Windwalker, is most often referred to by their species name, but the three heads were named individually, probably by Termagant. Stormfly, Horrorcow, Bullheart, Hookfang, One Eye, and many other dragons were named by the humans who kept them. They don't seem to have or seek other names because, well, why would they? Names are a human thing. Most dragons never saw a need for them in the first place.

The Windwalker has an intensely negative history with humans when Hiccup finds them. And so it makes sense that although Hiccup claims them as his riding dragon, he doesn't take ownership of them or force a connection with them or impose human customs on them by naming them. If the Windwalker wants a name, they can choose one for themself or ask Hiccup to give them one. If they don't want a name, they don't have to have one.

It's also possible Hiccup is taking a cue from his mother here - Valhallarama has a close bond with her riding dragon, an entirely unnamed Silver Phantom. He might've learned from her that names aren't necessary for friendship.

*(Exceptions do apply - the Green Death (who went by Merciless in the time of Hiccup I), the Purple Death (appears briefly in book 1), and Luna (the second in command of the dragon rebellion), all have names but don't have significant connections to humans, and all are members of the species Seadragonus Giganticus Maximus. I think it's likely that Furious chose his own name, but with the Wodensfang it's hard to tell - it's possible he was given the name by Hiccup I, but it's just as likely he picked it himself, or grew into the title as he aged. There definitely seems to be a custom of names/titles that's unique to this dragon species, though.)

YEAH

Hiccup: