How to tell if old ceiling light box can support ceiling fan?
The threaded rod you see in the middle is a stud. It is used to  attach mounting bars or hickey (a connecting device). Studs are attached to the box, not to framing members. It is no indicator of the strength of the box itself, or the attachment mechanism.

Fan boxes are characterized by their screw system that ensures a redundant attachment between the fan and the box. They also must be solidly mounted to framing members.

There are no heavy screws showing in the interior of the box, and the screw in the clamp holding the NM wiring going straight up suggests that there is not a framing member directly above the box.

It is both against code, and more importantly, dangerous to mount a fan in an unapproved box. The weight an dynamic load of a fan makes failure a high risk.

You need to replace the box with an approved box and ensure that the box is firmly attached to framing members. It may require some drywall surgery, but it is worth it.