The faculty of language is thought to be uniquely human. Recently, it has been claimed that songbirds are able to associate meaning with sound, comparable to the way that humans do. In human language, the meaning of expressions (semantics) is dependent on a mind-internal hierarchical structure (syntax). Meaning is associated with structure through the principle of compositionality, whereby the meaning of a complex expression is a function of the meaning of its constituent parts and the mode of composition. We argue that while recent experimental findings on songbird call sequences offer exciting novel insights into animal communication, despite claims to the contrary, they are quite unlike what we find in human language. There are indeed remarkable behavioral and neural parallels in auditory-vocal imitation learning between songbirds and human infants that are absent in our closest evolutionary relatives, the great apes. But so far, there is no convincing evidence of syntax-determined meaning in nonhuman animals.