The Old Speech is not like other languages in the world of Earthsea. In "A Description of Earthsea", the last chapter of Tales from Earthsea (2001), Ursula Le Guin also calls it the True Speech and the Language of the Making ("with which Segoy created the islands of Earthsea at the beginning of time").
Here are a few other comments on the Old Speech from "A Description of Earthsea":
Dragons are born knowing the True Speech, or, as Ged put it, "the dragon and the speech of the dragon are one."
Sorcerers and witches learn a few words of it; wizards learn many, and some come to speak it almost as fluently as dragons do.
Old Speech is also the language of spells; none of the other languages (Hardic, Kargish and Osskili) serve for the making of spells.
The Old Speech is written using True Runes (at least by humans; dragons have no writing). About lying, Le Guin write (my emphasis):
Human beings cannot lie in that language. Dragons can; or so the dragons say; and if they are lying, does that not prove that what they say is true?
So Le Guin does not really explain why humans cannot lie in True Speech, and the dragons' claim that they can has apparently not been verified by humans. This seems to be why Le Guin resorts to a variation on the liar's paradox (see the bolded text in the above quote). If a dragon were to say, "I am lying", and he/she is indeed lying, then the dragon is telling the truth, which means that it is actually lying. (The problem here is that the statement works both at the level of "object language" and at the level of "meta language"; one might argue that the two levels should be kept apart.)
Le Guin plays a logical game with the claim instead of explaining whether the dragons can lie or not.
Update in response to a comment: The quote from A Wizard of Earthsea, may look like a direct statement that the dragons can lie, but I consider that as written from the point of view that is inside the Earthsea universe, while "A Description of Earthsea" is written by an outside observer of that same universe. This outside view can be seen in statements such as "while the Hardic runes, like Chinese characters, can accommodate widely varying pronunciations and shifts of meaning" (in the section "Literature and the Sources of History").