Weight Paint: you cannot select a bone if either Paint Mask or Vertex Selection is active.
Since they look similar to vertex and face selection in good old Edit Mode, once one of them was active (not on purpose) I was not expecting that I can deactivate both at the same time, rather than always having one of them active.
Vertex Weights: this panel is only added to the vertex info (which looks the same otherwise) if you click on the vertex, not if you create a selection including only the vertex, e.g., by box-selecting over it.
(Concerns Blender Animation & Rigging course, Lecture Rigging & Weight Painting a Leg, from around time 6:16. I am using Blender 3.6.5 on Windows. )