Color doesn't have actual temperature. Try putting a blue square and a red square up on your monitor and hold a thermometer against both regions. If you find that there's a difference, you're doing it wrong. You probably know this already.
So why is color temperature measured in Kelvin? Kelvin is a measurement of the heat in a substance from absolute zero. That means, when there is actually no heat whatsoever in a substance and the molecules in it are absolutely still, that's 0 K. 0 K may not actually be possible, but that doesn't stop us from measuring relative to it, and this is a digression anyhow.
Is there some substance that emits different colors at different temperatures, which has been used as a reference to map temperature to color temperature? Or is it more complex than that? Or is the choice to use Kelvin completely arbitrary, with no relation to heat at all?
Answer
It is related to a heated substance, albeit in a somewhat theoretical way. The substance is an ideal incandescent black body, which would radiate a given color within a given color space at a given temperature. The location within the color space vs. temperature is called the Planckian locus, and I don't claim to understand everything in that article, but explore it to whatever depth you'd like.
For a more general "light reading" explanation of color temperature and it's correlation to black body radiators, see Wikipedia's Color temperature article.
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