Thanks for this comment, it was enlightening. I'd never much thought about this because physics is really not my forte, but after a few minutes of thinking about it, it's really intuitive why a point singularity can't spin. I can't explain it in any properly rigorous or technical way, but it makes no sense that someone could (by convention, of course) designate a "front" and "rear" to a mass with zero length in every direction.

What does it mean “to spin” in this case?

It means the black hole has angular momentum, which is detectable due to the resultant frame dragging (effect of the spinning on the spacetime around it) and presumably also due to the flattening of the sphere of the event horizon.