Why can two cameras give different brightness at the same ISO, shutter speed, and aperture?

You're right to be skeptical of ISO sensitivity. Yes it is a standardized, but the standard effectively allows for manufacturers to specify their own sensitivity. For a long time ISO320 on a Nikon was equivalent to about ISO400 on a Canon.

But that difference, whilst noticeable, was only a third of a stop. I wouldn't put a whole stop down to ISO differences. It would be silly for a manufacturer to label their camera as ISO3200 when it's the same sensitivity as another manufactuer's ISO1600, that way it will just make their camera look more noisy, as people compare cameras at the same ISO number, not the same actual sensitivity!

I think it's more likely to be a difference in tonecurves applied during raw processing that makes one image seem darker.

Image processing is the weak link, where it can all break down. Shutter speed is measured in seconds, a one second exposure on one camera is the same duration as a one second exposure on another (ignoring relativistic effects!). Ok there are tolerances so the duration wont be exactly the same, but the point is there is a unit, which is well defined. Same with aperture. But there is no unit of brightness in an image.

ISO [badly] defines how much actual light (a proper measurable quantity) it takes to saturate a sensor. But it's the image processing software that translates that into the numbers that get displayed on your screen. Even if the software maps a fully saturated sensor to the value 255 (which would seem to be the only sensible thing for it to do), a half saturated value might also get mapped to 255 or any other value for that matter. And that mapping can dramatically affect the perceived brightness for an image.