Should shutter speed match frame rate when recording video with a Nikon Z6? Or any camera for that matter?

Should shutter speed match frame rate when recording video with a Nikon Z6? Or any camera for that matter?

No.

As an example you might record at 60fps. So the longest exposure for each frame can only be (a tiny bit less than) 1/60th of a second. But if there's plenty of light, or you want to record sharp images with no movement blur, you can choose an exposure per frame of 1/1000th of a second.

All the frame rate determines is the slowest shutter speed, not the fastest one. So you couldn't have a 1/30th exposure at 60fps, but you could at 30fps. It also means if you want to record very high fps to achieve slow motion replay, then at 1000fps your maximum exposure is 1/1000th and unless you have a very fast lens (very large aperture) you also need a lot of light.