Proverbial can also indicate a well-known or familiar way of saying something, so your are correct in saying this is the proverbial eye of the storm, since it's a metaphor we commonly use to describe the phenomenon
Let’s see if I can explain. “The Eye of the Storm” is a proverb when you’re talking about a lull in a non-storm disaster. When you’re talking about a literal typhoon, that phrase is just the actual name of the weather pattern.
So the “eye of the storm” we’re talking about is the concept and origin of the phrase that is widely used as a proverb for non-weather related problems. Does that make any sense?