[D]olphin signature whistles are exclusive to individuals, rather than being part of a shared repertoire. They’re social sounds, unlike bird songs which are largely used to attract mates or defend territories. And they’re learned; many animals like birds and monkeys use distinctive calls to refer to specific objects (like different types of predators), but these are innate and inherited behaviours. “The use of new or learned sounds to label an object or class of objects is rare in the animal kingdom,” says King.
This combination of traits makes the signature whistles unique in the non-human world, at least for now. It’s possible that parrots and other birds might use similar calls.Reference: King & Janik. 2013. Bottlenose dolphins can use learned vocal labels to address each other. PNAS http://dx.doi.org/10.1073/pnas.1304459110
The fact that so much of the conversations of dolphins consist of proper names also suggests that while dolphins do have a more human-like language than most other species of animals that their language probably has considerably less lexical content (i.e. fewer words) than even the most primitive human language. A dictionary of dolphinese would have a huge directory of individuals with only a modest vocabulary beyond that. The way that the signature whistles are used, however, suggest that there is a fair amount of "grammatical context" to what one means and when one uses one based on context and intended meaning.