This is the sort of funny conversation one can imagine having spontaneously. But I can't help but point out that 'A simile is like a metaphor' is not a simile; not every comparison using 'like' is a simile. Similes and metaphors are actually alike in that they are figures based on comparison of two clearly unlike things, e.g., "The clouds are like sheep grazing lazily" and "The sky's sheep grazed lazily." When you are comparing things precisely as similar, you don't have a simile; it's only a simile when you are comparing them as similar at a point where they are in some way dissimilar. "Tom is like his twin brother" is not a simile. If I say you are clever like a fox, this is not saying that you are cleverness is actually the same as that of a fox; it is saying that the proportion of your cleverness to being human (if that's what you are) is itself proportional to the proportion the cleverness attributed to the fox has to being a fox. If I say you are swift as a deer, this does not mean that you have the same kind of swiftness a deer has, which actually makes you like a deer in that respect, but that your swiftness is to you like the deer's swiftness is to it. But similes and metaphors really are alike; they are based on the same kinds of proportions or analogies, they are figures of speech, and you can go back and forth between them while barely noticing (e.g., "The clouds, like the sky's sheep, grazed lazily"). The only point at which they seriously differ , even at superficial glance, is that one explicitly compares and the other simply substitutes or identifies.
But when we look less superficially, we see that similes are metaphors; this is the only way they can remain forms of figurative language. They are metaphors where the metaphorical expression is itself comparative. That is, if you take a metaphor, e.g.,
You are a deer
The corresponding simile is also a metaphor, but where the identification is not to 'a deer' but to 'a thing deer-like in swiftness',
You are swift as a deer.
But if we made this comparison simply by stopwatch, it would not be a simile at all; it would be a mathematical equality, and no one thinks that mathematical equations are similes. Similes must be figurative; bare comparison is not enough. Only if it is still a metaphor using 'deer-like' rather than 'deer' do we actually have a simile. This does make similes difficult to distinguish from ordinary comparisons; the comparative element makes the metaphor much more vague than it otherwise would be, which can mean that it can at times be hard to distinguish from a flat literal comparison. This is perhaps why, in comparison with other kinds of metaphor, similes are often given more extensive explanations when they are actually used: we often need to compensate for the potentially confusing vagueness introduced by the comparative element.
But 'analogies are like sandwiches' is very much a simile.
