As you mentioned before, あおい refers to blue as an adjective, and あお as a noun "the color blue." Addressing your question directly, あおい as an adjective is referring to the property of an object that is color.
If you think of an adjective (in this case, an い-adjective that is あおい) as a word that tells you about a property of an object, you might be able to tell that there is some usage overlap that can give you a similar meaning between the adjective and the noun counterpart for this word. However, a clear distinction needs to be made that is of the nature:
X(noun) は Y(noun) です
X is Y. (X is equivalent to Y.)
X(noun) は Y(adj.) です
X has a property which is Y.
It is possible to state the following with both words:
This ball is blue.
...However, the meaning is slightly different between the two:
This ball is blue. (lit. is of a blue nature, has a blue "color" property, etc.)
This ball is blue. (lit. this ball is LITERALLY "the color blue").
As for using the color あお as denoting the "blue" feeling, this is a concept that is not native to Japanese. However, you could use ブルー "blue" to say this. This is not something that everyone will understand universally, but is present in at least some form of popular media (the song 仲良し by Spitz comes to mind for this). The lyric that contains this usage is: