I don't see this as a major issue, nor do I believe that the word "unique" is in any particular need of saving. There are a large number of terms in mathematics which correspond to vernacular English words, but which have distinct technical meaning in mathematics. For example, an "odd" number is an integer which is not a multiple of $2$, rather than a number which is in some way strange or peculiar (though, oddly enough, the mathematical definition of "odd" predates the modern vernacular usage). As another example, in mathematics "onto" is a synonym of "surjective", while in vernacular English "onto" is a preposition which is of general utility. Or, as a particularly perverse example, what does "normal" mean (hint: even in mathematics, this little word has a lot of different meanings, depending on context)?
Note, also, that this isn't an issue unique (heh heh) to mathematics: in archaeology (and geology, maybe?), the word "flint" refers to a specific type of toolstone which is associate with limestone deposites; in vernacular English, many types of crypto- and microcrystalline silicates are referred to as "flint".
One of the jobs of an educator is to introduce their students to the technical jargon of their field, and to help students to understand that words may have a precise definition which is different from the vernacular meaning, or different from the technical meaning in another field (indeed, "unique visitors" is a well-understood term and has its own technical meaning).
In the example question and answer posed above, I would regard it as an opportunity to discuss this distinction between vernacular usage and mathematical usage. For example, on an exam, I might write something like the following:
Question: Define the term "one-to-one" for a function $f$.
Answer: Every $x$ has a unique $f(x)$.
Response: I understand what you mean by this answer, but this is not the correct usage of "unique" in mathematics. It would be better to say "Every $x$ has a different $f(x)$," or even better to say "Every $x$ is mapped to (or sent to) a distinct value by $f$."