Why every claimed human-only feature keeps turning up in animals
Scholars have repeatedly proposed properties that supposedly make human language unique—things like reference (signals that stand for objects or events) or syntax (rules for combining signals). Each time, researchers have found some attenuated version of that property in animals, which is why the uniqueness argument has gradually shifted from linguistic rules to the cognitive capacities underneath them [2]. In other words, the debate is no longer simply about whether animals have words or grammar, but about whether they think in the ways that make human language possible.
One current proposal is that human language depends on ostension and inference—roughly, pointing at things and drawing conclusions about what a communicator intends—while animal communication is characterized as simple association and largely hardwired signals [2]. That contrast does show up in published data, but the same paper cautions that the findings are shaped by radically different methods used to study animals and humans, so the contrast may be partly an artifact of how each is tested [2].
Language evolution may depend on animal minds, not just animal signals
Comparing animal communication with human language is not enough on its own, because animal cognition has an important role in understanding how human language evolved [3]. The fact that animals have cognitive abilities that overlap with those involved in language means that focusing solely on communication signals could miss part of the picture [3].
An overview of the research likewise compares the components of human language with non-human animal communication systems and examines continuity and differences between them [4]. Taken together, these comparisons suggest that the question is not simply whether animals have language-like signals, but how much of the underlying cognitive machinery is shared.
What a fair comparison between animal and human language would need
A comparative psychology of language argues that the study of language—whether human or animal—is fundamentally the study of the same phenomenon, and that what we know about the nature and functions of human language is relevant to animal language research as well [5]. That framing supports treating the two as comparable rather than categorically separate.
The neurobiological and evolutionary approach adds that animal models can reveal the neural mechanisms and genetic adaptations that underlie human language [1]. So the most productive comparison may combine careful attention to method—avoiding biases built into the code model [2]—with attention to cognition and brain mechanisms [1][3], rather than relying on teaching attempts alone.
Sources used in this answer
What can animal communication teach us about human language?
A comparative neurobiological approach using animal models can reveal neural mechanisms and genetic adaptations underlying human language.
Overcoming bias in the comparison of human language and animal communication
Biases from the "code model" and different study methods shape comparisons of animal communication and human language, and breaking away from code-based approaches may show differences of degree rather than kind.
Animal cognition and the evolution of human language: why we cannot focus solely on communication
Animal cognition plays an important role in understanding human language evolution, so focusing solely on communication is insufficient.
Animal communication and human language: An overview
An overview compares components of human language with non-human animal communication systems and examines continuity and differences between them.
Similarities and differences in human and animal language research: Toward a comparative psychology of language
A comparative psychology of language treats human and animal language as the study of the same phenomenon, making knowledge about human language relevant to animal language research.