It is just an idiom and has no history.
spying or looking someone all the time
The idiom "keep an eye out for" refers to watching for something or someone. An example of a sentence using the idiom would be: Jeff should be arriving soon, so keep an eye out for him.
Keep watch on.
it is an old military term for good sight
The image is of you pointing an eye at something or someone. It means to watch carefully.
An excellent expression. To keep an eye, or to keep an eye on things means to maintain a watchful eye. You monitor what is happening with an appropriate zeal to the occasion. Another idiom would be, to keep an eye peeled. It means you are watching and protecting something, or someone.
The earliest recorded use in Modern English is in Sir Walter Scott's Old Mortality, 1816: "Poor Richard was to me as an eldest son, the apple of my eye."
To be exposed
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Origin "up a storm"
No, the idiom is "turned a blind eye" as in "she turned a blind eye to his suffering." You don't change an idiom around or you lose the meaning.