Does a fan need a flyback diode?

It's a brushless DC motor. Most, if not all of them, have flyback diodes across the windings (inductors) which are switched electronically with transistors in the motor (contrast to mechanical commutators in brushed motors).

There shouldn't be any need for a flyback diode across this fan, unless there is some significant series inductance eg a EMI filter inside the motor.

But, when in doubt, add a flyback diode. It's cheaper to add one unnecessary diode than frying your circuit because you did need one.

A DC motor is a capacitive load. (at-least in the scale of seconds it is)

In the scale of microseconds it may appear as an inductive load.

The type of fan pictured has internal electronics that will not produce any inductive behaviour on the supply terminals.