Yes and no. It’s a circular relationship. The proverb is known to have already existed in Ancient Greece. The pantheon (the version that existed before Xerxes torched the place) was apparently inhabited by tons of owls, especially it’s roof construction. Since the pantheon was a temple to Athena, people assumed Athena held Owls holy and the owl became linked to Athena, and since Athena was the goddess of wisdom, Owls became a symbol of wisdom. Since the city and her patron goddess are related by name, Athens is linked to owls as well.

But for the meaning: It’s to be taken literally. There were many owls in Athens, so they’d not exactly need any more.

Originally it must be, but the animal itself became associated with the city too (for example their coinage almost always had an owl on it)

I don’t know. My coworker just said that owls are a symbol of wisdom, and Athens (historically) is the center of wisdom.