I was actually under the impression when you wrote a mathematical formula without stating that x has a particular (but omitted) value, it usually more formally is a stand-in for when you do decide to supply a value.
In other words, I do not know if a “free variable” is a concept in more axiomatic set theory. It’s more of a notational convention that’s saying, “for any x in this domain, this function associates it with the following expression”. So actually, the “free variable” is not conceptually different from the “bound” variable. They are only symbolic placeholders to be substituted with a value.
That said, we can think more philosophically about what you are saying. I am pretty sure it could be the pretty well-established idea of “sign” vs. “referent”. In your example, x is a “sign”. Once you state what it means, now the sign has a referent. This is a general characteristic of language, in that we use conventions and associate them with meanings.
Maybe another way of thinking about it is that it is possible to talk about a thing not merely without specifying which thing it is, but far more abstractly, that it is not even asserted what kind of thing it is, yet some proposition with regards to it is still coherent. I think this could be like saying “one of the marbles in the bowl is green”, vs. “Things which are heavy sink fast in water.” This may not be a perfect explanation, but the idea is indicating very clearly what set of things something is in and making it clear that you have decided mentally which one you are referring to, vs. talking about some hypothetical thing that technically has not been claimed to be real, yet. And, honestly, this I believe is part and parcel of semantics. I believe it’s called bound vs. unbound variables and has to do with the field of pragmatics, concerned with how people infer meaning from context, such as what an anaphoric word like “it” is referring to.
(P.s. linguists use variables.)