That rating on the power supply shows that it is running the lamp.
The motor beside the lamp looks like it is a synchronous motor, which need the AC output from the power supply. All laptop power supplies that I have come across have a DC output, so they won't run the motor, but would run the light OK if the votlage and power rating were near enough.
If the lamp is working on the power supply shown, then the power supply is working fine.
Motors like that are fairly simple, and don't have brushes. I can't see what the post is. Can you get a better photo?