Is there ‘syntax’ in animal vocalizations? Maybe/it depends what you mean. In any case there sure are some complex long timescale dependencies that look rule-like! The bird song people are making a lot of progress on understanding how this structure arises.
I think progress here requires a few things: 1) better conceptual clarity across the board and 2) large scale lab studies where we can monitor vocalizations and social interactions over months/yrs. This will allow for much more fine grained tests of our theories of animal communication.