The title, basically. I always understood that the natural, harmonic and melodic minor scales were 3 different scales deriving from the same "minor" key. I hear people talking about a melodic minor "key" when referencing the modes of the melodic minor scale, but this sounds incorrect to me. Can the melodic minor "key" exist?
Viewpoint #1: melodic minor is a scale, not a key
More specifically, it's a scale constructed as an abstraction of how composers use minor keys in actual practice. Many composers used the raised sixth and seventh when a passage moved upward toward the tonic, but the lowered sixth and seventh when moving away from it.
A piece composed strictly according to the conventions of the melodic minor scale, would still be considered to be in "X minor", not "X melodic minor".
The same is true of harmonic minor: it abstracts another approach composers have used when writing in the minor mode.
Viewpoint #2: melodic minor is (i.e., can be treated as) a mode
(Along the lines of Andy Bonner's cogent comment that being in "the key of X melodic minor" is a shorthand for being in "the key of X minor, and [using] melodic minor behaviors" ...)
When we speak of being "in a key", what we are usually referring to is being "in a key and mode". So "in the key of X" means "X is the tonal center" (the key) and "major is the mode" (the selection of notes available). Similarly being in "X minor" means "X is the tonal center, and minor is the mode."
In this way, especially when composing using a mode of the melodic minor scale, it might be convenient to say the piece is in "the key of X melodic minor's third mode", and so forth.
Thus: if one interprets melodic minor as a mode (a set of notes with certain defined compositional conventions), then it could be considered correct to say that a piece is "in melodic minor".
In these days when a piece is as likely to be in a mode as in a key, terminology sometimes gets a bit loose! But you're right, C minor is a key, C melodic minor, C harmonic minor and C Natural minor (the one that goes both up and down keeping to the key signature) are scales that are used in that key.
Its been realised, over time, that certain notes work well in certain keys. More so with the minors. It's interesting that minor particularly involves the m3 from root. That is the quintessential component of minor. All minor keys and modes possess that m3.
Any other notes which are used often in 'minor' pieces will contain many different notes, for many reasons, reflected in the various minor scales. Scales are basically a specific set of notes (which work together) set out in order.
So, a piece in a minor key will use m3 rather than M3, but all other notes are usable - and used. There are no 'rules' as to which notes are used - apart from the age-old raised leading note , therefore raised 6th note also, so in a minor piece, many notes are used. That doesn't mean a piece is in harmonic, natural, or melodic minor, but simply in minor.