What's wrong with these stairs?

[Bret Habura]

Hello! I just had an inspection done on a 2 unit home yesterday. During the walkthrough we noticed some issues with two of the staircases (pictured below). When you push on the red circle, there is noticeable movement. Obviously this isn't what you'd expect, but I'm frankly not sure what the risk is or how it would be fixed. When you walk up the stairs there wasn't any movement.

The inspection report reads: "Many older homes have stairs that don't meet current requirements for size of tread, height of steps, or head room clearance, guard rails and hand rails. These stairs can be a trip/fall hazard. The stairs in this house have some or all of these concerns."

Has anyone seen something like this before?