Noise from floppy drives is dominated by seek noises (as demonstrated in the video you linked to), to the extent that in normal operation you don’t really notice the spinning noise.
In practice, regular floppy use tends to be mostly short bursts of this kind of noise (three seconds’ worth starting at 1:07). Formatting a floppy or reading/writing it in its entirety sound different, and are also immediately recognisable. You’d learn the noises your drives made, and quickly notice when something was wrong; but most of the time, “when something was wrong” was when a sector could no longer be read or written, and the drive would repeatedly try, resulting in this sort of noise or this sort.
Unusual spinning sounds were rather unusual and would indicate something badly wrong, either with the drive itself, or more commonly with the floppy — in my experience the most common “wrong” spinning sounds came from slightly bent casings, so the disk would rub.
Spinning sounds involving the head tended to be bad news. A scratching sound would mean the floppy was toast, and if you were unlucky the drive as well; or if you weren’t paying attention, the next floppy you put in the drive. (Dirty floppies leave deposits on the head, which then damages the next floppy if it isn’t cleaned.)
When floppy drives were still common, this was all true of hard drives as well. You would know the noise patterns your computer would make when booting for example, or when loading programs you used often, and you’d quickly notice any change.
