She looked me in the eye

I understand "she looked me in the eye" is a set phrase, but I wonder why not "in the eyes." Can anyone explain?

In a practical sense, it's actually physically impossible to look someone in both eyes simultaneously--you would have to focus on one or the other, or else switch back and forth between them. However, there is also a general linguistic pattern at work here. We say "I couldn't look them in the eye", even though this case definitely involves multiple eyes (although still not simultaneously).

According to Google ngrams, "look [someone] in the eyes" is still relatively common, although less common than "look [someone] in the eye". Expressions like "easy on the eye/eyes" and "easy on the ear/ears" also tend to have singular vs plural used interchangeably, although one or the other will usually be more common. In expressions like these, the body parts in question can serve as a type of metonymy--they are used as symbols for something abstract that is somehow associated with them. But it's also possible to take the expression more literally. For example, "He couldn't look them in the eyes" can mean that he was generally unable to confront them, or that he did confront them, but could not look them in the face. Using the singular "eye" resolves the ambiguity by making it clear that the body part is definitely being used as a metonymy.

Here are some more examples of how the singular is used to highlight the metonymic nature of the phrase:

"This color combination is easy on the eyes": could be taken literally (colors are not jarring) or figuratively (colors are esthetically pleasing).

"This color combination is easy on the eye": more likely to be understood figuratively.

"This sound is easy on the ears": is it literally a sound that doesn't hurt the ears, or a good melody?

"This sound is easy on the ear": more likely to be understood figuratively.

"She was tired of his roving eyes": does that mean him specifically checking people out, or just sexual interest in other people in general?

"She was tired of his roving eye": definitely refers to general interest.

"his iron fists": most likely refers to punching ability

"his iron fist": most likely refers to ability to control things

"nature's red teeth and claws": could be a symbol for general mercilessness or a synecdoche referring to actual animals with teeth and claws

"nature, red in tooth and claw": clearly a symbol of general murderous tendency

Obviously, this method of resolving ambiguity is only available for items that are always expected to come in a set, such as eyes, ears, arms, etc. For example, if I say something like "the pain in his heart", and it's not clear if I mean literal chest pain or metaphorical hurt feelings, I would have to find some other way to resolve the ambiguity.