The language you speak of is Enochian, correct? It is accepted as such because of humans who interpreted a spiritual energy into a language, which can be done for dragons and other kinds of spirits except for the fact that there is no commonly known pantheon of dragons. Dragons from lore come from all over the world and while I cannot speak for them in having their own language or not, I can say that trying to interpret a universal language between spirits of multiple paths could be a difficult task.
Even though Enochian is considered the language of angels, you have to account for human error. We can't consciously comprehend the spiritual essence, but we can interpret them in a manner that we can understand. For all we know, Enochian in actuality can look like a blob of water. The same goes for a language of dragons or a language of any other kind of spirit.
I would begin researching the lore where dragons, serpents, and wyrms come from in their original context. For example, Fafnir is cited as a serpent or a dragon, but he is really a Dwarf (or Dvergr) that has changed his shape into one.