Can a phrase be both a metaphor and an idiom?

According to ODO, there is (even today) a literal usage for the compound noun rain check:

rain check NOUN

North American

1 A ticket given for later use when a sporting fixture or other outdoor event is interrupted or postponed by rain.

and of course this is where the broadened usage has come from:

take a rain check [phrase (in the broader sense of the word)]

Used to refuse an offer politely, with the implication that one may take it up at a later date.

Since the broadened usage is obviously not a literal one, this usage is a metaphor. Yourdictionary.com gives the most helpful definition of metaphor here:

'[a figure of speech where] a word or phrase ordinarily and primarily used of one thing is applied to another'

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The term 'idiom' has been defined in a multitude of ways {see Fixed Expressions and Idioms in English ... Rosamund Moon for a thorough overview of this problem}. I prefer the definition 'a multi-word string reasonably commonly used and considered acceptable by proficient speakers, using otherwise non-standard grammar and/or an otherwise unknown sense of a word'. Notice that this definition doesn't mention degrees of transparency / opaqueness.

Since 'take a rain check' (cf 'take a sun hat') uses neither unusual senses of any words (one might argue somewhat old-fashioned, but that's different) nor unusual syntax, I'd maintain that this is not an idiom. Though it's certainly a fixed expression.

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'She led him a merry dance' is certainly a metaphor and arguably an extra-grammatical idiom; I've seen an argument for ditransitivity here, but it is extremely unconvincing.

Idioms, metaphors, and similes are all oddly similar things. And the latter two are what we would call "figures of speech".

The main difference between an idiom and the others is that an idiom has become popular in a native language. It's known to the listener whether they know they meaning of the individual words, or the phrasing, or not. They know it because they have heard it so much they know what it means regardless of the individual words or meanings or definitions.

"Figures of speech", which include metaphors and similes, are really only different in that their meaning can be determined even if the listener is not familiar with the phrasing. What does that mean? Okay, for example, let's say you know what, "the early bird gets the worm", means because you have heard it all your life. You understand it automatically without even considering the individual words. Now, let's take something like, "necessity is the mother of invention". You may never have heard this before. However, because of the descriptive language you are able to deduce that this means necessity is what moves or progresses invention. That is the real difference between the two.

So, from these standpoints we could say that all metaphors and similes are idioms but not all idioms are metaphors or similes.

However, the bottom line is that they are all extremely similar, and for all intents and purposes, in my book (another metaphor, or idiom, depending on point of view ;) ), could be very nearly considered synonyms of each other.

Short answer, yes, by definition.

An idiom is a phrase whose meaning cannot be established from the combination of its individual words, usually by repeated use in other contexts.

A metaphor, or more generally a figure of speech, is a nonliteral way of understanding a phrase (for metaphor, by analogy).

An idiom is non-literal and a figure of speech is non-literal, though their emphases are different. An idiom is opaque but a figure of speech is more poetic. A particular phrase that uses any one of the strategies of figure of speech (metaphor, synecdoche, personification, etc), can become an idiom by overuse.

In particular, a metaphor that has become a dead metaphor. Their example, 'Time is running out' is a metaphor because time can't literally run but it can feeling like it is flowing quickly along like someone running. It's also an idiom because no one (native speaker) has any inkling about flowing when they say it, it just means immediately that there is no more time.

Of course other figures of speech could be considered idioms. 'White House', used as a stand-in for 'the presidency of the USA', is metonymy. But it is also an idiom because you'd never know it was referring to the presidency unless you knew the name of the president's residence.

I can understand the idea that an idiom is relevant to a specific region and therefore readily understood by the people in that region. We used to use the term submerged or implied metaphor for nouns, adjectives, or verbs that are being used in an unusual way. Example: The baseball team crushed it's opponents.