These type of boxes usually have 1 or 2 screws going through the bottom into a steel bar that spans between ceiling joists.
I think you are correct that this box is not properly supported.
If you have access from above, you can put a support (2x lumber) across to each joist and then screw through the box into the support.
Options from below only would involve removing the existing box and using one of several replacement boxes. Removing the existing box will involve turning off the power and prying the electrical cable loose from its strain relief. From the looks of the box, it should pull down. Fixture weight will determine which one to use.
The heaviest duty box is one rated for ceiling fans. It uses expandable threaded supports that wedge against the joists side to side. This box would be rated up to 50lbs.
An old work round plastic box has three retractable wings that clamp the box against the drywall. This box is suitable for lighter fixtures, perhaps up to 10lbs. It is very dependant on intact and unbroken drywall for best performance.