Researchers have found that songbirds have something that resembles grammar as we know and are very responsive to rule violation. The birds have a syntax in their tweets, maybe not the same concepts like us (that is nouns, pronouns, verbs, adjectives, adverbs and so on), but they have a syntactic structure. Syntax is the study of principles and rules for constructing sentences and grammar rules are a part of syntax.
Language is made up of signs, meanings and a code connecting signs with their meanings. Semiotics is the study that looks at how signs and meanings are combined, used and interpreted ( you can read it up in our paper Semiotics and Intrinsic Semantics).