"Very" is an intensifier. Normally it is used as an adverb modifying an adjective or another adverb. "Bob is very tall." Not only is Bob tall, but we want to emphasize how tall he is. "Sally drives very recklessly." Not only does she drive recklessly, but we want to emphasize how reckless.
There is a relatively rare usage where "very" modifies a noun, like your example. In this case it means the extreme or most stereotypical version of that thing. For example, "I am the very model of a modern major-general." Not only is the speaker claiming to be a model of just what a major-general should be, but he's "the very model", the most perfect example one could imagine.
Or in your first example, not only is he sophisticated, but he is the "image of sophistication", the very idea that comes to your mind when you hear the word "sophistication". And more than that, he's "the very image", the most stereotypical or classical example.
Usually, I think, when "very" is used with a noun, the noun is a word indicating a class or stereotype. We say, "the very model of a major-general", not "the very major-general"; or "the very image of sophistication", not "the very sophistication".
"Very" can also be used as an adjective to emphasize that you mean that one specific thing. Like, "That rope was the very thing we needed to escape." Any old thing would not have done, we needed one specific thing, namely, that rope. Or, "This is the very town where I was born." This means pretty much the same thing as "This is the town where I was born", but I am emphasizing that it was this town and not some other town.
Your second example is awkwardly worded. I'm not sure if you were quoting someone or trying to create your own sentence from examples you've seen. A better wording would be, "This is the very idea of a story of a story" or "This is the very model ..." etc. You could also say, "This is very much the story of a story", but then that's a different usage, then you're using the more conventional form of "very" as an adverb modifying the adjective "much", rather than modifying a noun.