Is neutral wire being switched?

I’m replacing a light fixture with a ceiling fan and there’s a black, white, and ground wire in the ceiling box (each branched off of a cluster of wires that continue on to power the wall outlets). But when I turn the wall switch off, the black wire still is detected as live by my multi-meter. Note that the wall outlets are always on, regardless of whether the wall switch is on or off, but the light would turn on and off with the switch.

The multimeter measures no voltage between the black and white when the switch is off (but does measure voltage when when it’s on).

I don’t know much about wiring, but does this mean the neutral wire is being switched, and is it dangerous to hook up the fan anyway?