In my understanding-- the analogy is the entire comparison, and an analogy can be made by employing a metaphor, which is a specific figure of speech-- like, a ladder being a symbol for upward mobility would be the metaphor, with the entire phrase/story in which this takes place being the analogy.

Maybe?

And at least in my classes we learned that metaphors are always presented as a direct swap (no "like") -- for example "all the world's a stage" is a metaphor, but if the quote were "the world is like a stage," that would be a simile.