A practical explanation is that "Kyne" is a homophone with an existing dragon word, kein "war", so she has a unique name in the language to avoid possible confusion there.

We don't know whether or not other gods have unique names in the language because they simply never occur in the available text or dialogue. Some place names like Atmora and Oblivion are borrowed directly into the language, so it's possible not every deity has a unique dragon language name.