Is there a good reason that my camera, a D5500, can't use the exposure meter with a manual lens attached?

Not really. All Canon dSLR bodies are capable of stop-down metering when they fail to sense electronic communication from a lens and the higher-end prosumer Nikon bodies can do accurate metering with an adapted lens. But the D3x00 and D5x00 bodies have metering systems that are only set up to perform wide-open metering. And wide-open metering does have the benefit of giving you the most light to compose/see by when using the camera, regardless of the aperture you've set.

... why can it not just measure what it can see?

Actually, the problem isn't that it's not measuring what it can see; it's that it's making the assumption the lens is wide open while doing so. If you are using any aperture setting smaller than the maximum aperture, then the metering system has to compensate for that. And Nikon simply didn't program that into their entry-level bodies.

Does this also apply to higher-end bodies, or is it true for any manual lens on any body?

No, it only applies to the entry-level bodies. The prosumer models are capable of accurate metering with non-CPU lenses, and some can be programmed to know the focal length and max aperture of such lenses.