A 'metaphor' is a figure of speech that refers to one thing by mentioning another.
Yes, it is a Greek word, meta-phoros -- "meta" is "beyond" and "phoros" is "carrying", the extending (carrying) of a meaning beyond its usual context.
It can apply to words, or to other media. One thing can be implied by
representing it with another. There is such a thing as a
visual metaphor. We use them constantly in our "Graphical User Interfaces" on digital devices. The concept of metaphor has long since been 'carried beyond' the sense of mere words.
Merriam-Webster has "an object, activity, or idea treated as a metaphor" as meaning #2. (
https://www.merriam-webster.com/dictionary/metaphor)
Since Sauron is
not physically a huge eye, it would need a pretty abstruse point to make the difference between lexical and visual metaphors important enough to call Frodo's vision something other than a metaphor.
Yes, Forodan, The Eye could be a metaphor. But it also might not. (Which is why I said it is not exactly a metaphor). If the Eye is symbolic of something else, for example, symbolic of Sauron, which it might be when used on the shields of Sauron's orcs, then it might be a metaphor.
However, this is not clear. When Frodo sees the Eye in the Mirror of Galadriel, is this a metaphor for Sauron? Is this actually Sauron's real eye? (We assume he has eyes, regardless of what his entire form might be.) (Also, there is not - that I can recall - in TLOTR, any suggestion that the Eye is huge.) Is this a metaphor for Saurons will? Is it a metaphor for Sauron's perception?
Really, a metaphor should be using something to describe something else. As in, 'he was the black sheep of the family', or as in a picture of a chain used as a visual metaphor for strength, or a strong connection. Using part of something, to describe the whole thing might not be exactly a metaphor. A picture of the head of a horse, I suggest, is not exactly a metaphor for the whole horse, even though the observer thinks 'horse'. Just like a passport photo of someone's face is not really a metaphor for the person. It is just a (partial) representation.
So, while it is possible that the Eye that Frodo sees is a metaphor for Sauron, it is also possible that it is just a partial representation of Sauron. Perhaps even a dramatized partial representation of Sauron (if we were to conjecture - on no evidence - that Saurons real eye would not be 'rimmed with fire'.)
I guess another way to consider whether the Eye is a metaphor, is to turn the visual into verbal and see if it flies? If we said, 'their love was an anchor chain', that would be a metaphor that compared one thing to another and made sense. If we said, 'Dave was an eye', would anyone know what that was supposed to mean? So, if we said, 'Sauron was an eye', would it make any more sense?
Not all symbolic representations of something else are metaphors. Suppose there was a knight, Sir Sydney Hightower, who's heraldic device on his shield, and on the surcoats of his retainers was a tall tower. Would that be a metaphor? Probably not. It would just be a visual literal interpretation of his name.
We guess that Sauron has eyes. We understand, from Pippin, that they have an impressive and powerful effect (the look). The fact that an eye is symbolic of Sauron for his orcs, does not necessitate it being a metaphor. It could be just representative of one of his distinguishing features.
So, I am not sure that the Eye is a metaphor.