Why does paint mix to produce black, but light mix to produce white?

Mixing light does result in white, but the black mixture of paint happens due to how paint works. Paint has color not because it's emitting light, but because it's absorbing colors other than the one that's supposed to be the paint's color. As such, when you mix paints, they absorb more and more of the spectrum, resulting in black.

Mixing light is additive since you are adding electromagnetic waves with different wavelengths together.

The color of materials as paint relies on a different principle. Only certain wavelengths of the impinging light get scattered back and the rest gets absorbed. For example, red paint only scatters back the red wavelengths and absorbs the others. (This also means if you light the red paint with blue light, you will not see much). Now if you mix all colors of paint, the whole (visible) spectrum of the impinging light will get absorbed. Hence, paint is subtractive. (This is also the reason why black paint gets warmer than white paint; the absorbed light is converted into heat energy).

It actually has nothing to do with light itself. An easy way to demonstrate this is to get some panes of differently colored glass, and line up a light source to shine through multiple panes. You'll notice the color gets remarkably darker the more differently-colored panes you use. Shining a light through a red pane, a green pane, and a blue pane won't get you white; it'll get you pretty close to black.

It's not light vs paint; it's light addition versus light subtraction.

When you see 'green paint', it's not because the paint is adding green light. It's because its subtracting out non-green light. Same thing with a pane of green tinted glass - it's not adding green light, it's subtracting out non-green.

Compare that to a computer monitor. If you've got a bunch of green pixels lit up, lighting up the neighboring red pixels doesn't subtract from the amount of green light; it simply adds additional red light to the equation - which your eye then perceives as a shade of yellow.

So, what happens when you mix a bunch of colors of paint? It actually doesn't become black - it just becomes a grayish mix matching the average brightness of the colors. Try it - mix Sky Blue, Canary Yellow, and Pink - you won't get a color anywhere close to black; but if you mix Navy Blue, Brown, and Forrest Green, you'll get something a lot darker. An easy way to mentally picture what you'll see is if you imagine painting a checker-like pattern out of the colors you're mixing, and then walk far enough away that the colors all blur together.