Aristotle is the source of our modern notion of univocity. He was also the first to point out many things of interest that overflow our attempts at univocal representation.
Something is said univocally (i.e., without equivocation) if the meaning is the same whenever the representation is the same. This is an important, desirable property in logic, modern science, and engineering. A lack of univocality is one of the major sources of logical inconsistency or incoherence in representation.
A kind of univocity also applies to a Kantian unity of apperception. The unity of a unity of apperception is more or less equivalent to the univocity of an account of something. So since there is a kind of moral imperative to achieve unity of apperception or improve upon it, there is in that way also an ethical use for univocity. But just as unity of apperception requires constant renewal, so too do our attempts at comprehensive univocal accounts of things.
Aristotle frequently points out things that are “said in many ways” (i.e., not univocally). This refers not just to practices of ordinary language, but to realities as well. Being and cause are among the things said in many ways. He also points out cases where a more fluid, pluralistic approach is appropriate — the classification of animals, for instance — or just applies such an approach. Aristotle wants to be faithful to the variety, subtlety, and complexity of the world and everything in it. This is the famous Aristotelian manysidedness, often praised by Hegel.
Univocity is never simply found; it is a possible property of our constructions and passive syntheses. In the design of representations and schemas, there may be tradeoffs between coherence and correspondence, or consistency and comprehensiveness.
If there is an ethical imperative to univocity, there is also one to manysidedness. These are really two sides of one coin (a sort of responsibility our representations have to reality, as Brandom would say). To paraphrase Whitehead, we should seek univocity and distrust it. (See also Aristotelian Semantics; Mutation of Meaning.)