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From Japanesepod101:
象は灰色をしている。
The elephant is gray.
The meaning of the sentence is not in doubt but I've been trying to figure how する is being used here. Checking a dictionary, definition 2エ might be applicable here.
ある状態・性質であることを示す。(To show something's state/nature)
It is exactly as you say.
「～～をする」 can mean "to have ~~ (as a natural characteristic)" and it is usually used in the form 「～～をしている」,「～～をしていた」 or 「～～をした」.
The "dictionary" form 「～～をする」 would not be used in a real-life situation; It is only found in dictionaries or a place like this where grammar or vocabulary is discussed.
Among those forms, 「～～をした」 is always followed immediately by a noun or noun phrase. You cannot end a sentence with 「～～をした」 when 「する」 is used for the meaning we are discussing. That is to say:
Always use 「している」 for the present tense and always use 「していた」 for the past.
している：「スミ子さんはきれいな[目]{め}をしている。」= "Sumiko has beautiful eyes."
していた：「[犯人]{はんにん}は[青]{あお}い目をしていた。」= "The culprit had blue eyes."
した：「[赤]{あか}い[髪]{かみ}をした[人]{ひと}が[好]{す}きです。」= "I like redheads." Or more literally, "I like people who have read hair."
