On "Key" vs. "Scale"
The phrasing of this question seems to be generating a lot of discussion about the difference between a key and a scale. First, I'll address that issue, and then I'll offer an answer with some references.
I think much of the discussion about key vs. scale is misguided because keys are based on scales. For example, Wikipedia states:
In music theory, the key of a piece is a group of pitches, or scale upon which a music composition is created in classical, Western art, and Western pop music.
If keys are based on scales, then a minor key must be based on a particular minor scale. So the question is valid in asking about the underlying scale on which the minor tonality is based. I see the question as being: is there a single minor scale that is implied when a song is stated to be in a "minor" key?
Answer: Natural Minor
The answer to this question is: in general, yes, a single minor scale is implied. Stating "E minor" implies E natural minor. For example, studybass.com states:
Similarly, a piece of music can be in a minor key and revolve around a natural minor scale. For example, a song in the ‘key of D minor’ uses the notes of the D minor scale – D, E, F, G, A, Bb, and C.
Furthermore, the same Wikipedia article from above states:
A key may be major or minor. Music can be described as being in the Dorian mode, or Phrygian, etc., and is thus usually thought of as in a specific mode rather than a key.
So even the Wikipedia article straightaway excludes Dorian minor as a key, implying that a different minor scale must serve as the basis for minor keys. This affirms the general position on studybass.com that a single minor scale is implied by a minor key.
This approach might seem to leave no room for keys based on other minor scales--the Wikipedia article states as much. However, there is still a way to describe a song written in a Dorian mode, but that description shouldn't include the word "key." A statement along the lines of "the song is in the key of E minor" implies E natural minor. Perhaps a song written in E melodic minor would be best described this way: "the song is written using the E melodic minor scale."