On the surface, the fields of animal communication and human linguistics have arrived at conflicting theories and conclusions with respect to the effect of social complexity on communicative complexity. For example, an increase in group size is argued to have opposite consequences on human versus animal communication systems: although an increase in human community size leads to some types of language simplification, an increase in animal group size leads to an increase in signal complexity. But do human and animal communication systems really show such a fundamental discrepancy? Our key message is that the tension between these two adjacent fields is the result of (a) a focus on different levels of analysis (namely, signal variation or grammar-like rules) and (b) an inconsistent use of terminology (namely, the terms “simple” and “complex”). By disentangling and clarifying these terms with respect to different measures of communicative complexity, we show that although animal and human communication systems indeed show some contradictory effects with respect to signal variability, they actually display essentially the same patterns with respect to grammar-like structure. This is despite the fact that the definitions of complexity and simplicity are actually aligned for signal variability, but diverge for grammatical structure. We conclude by advocating for the use of more objective and descriptive terms instead of terms such as “complexity,” which can be applied uniformly for human and animal communication systems—leading to comparable descriptions of findings across species and promoting a more productive dialogue between fields.