This thing is new to me because I didn't saw or noticed it (if it already exist on the older versions of Blender)( I'm using 3.01 ). If you go into Edit mode, select the Items tab, it will show you the Vertex Weights of a mesh.
In this mesh, you can see the Weights it has on the Items tab
In this one, it's doesn't show it
In the Numbers Panel you only see a weight of a single vertex - you need to have a vertex active (marked white). Since it's possible to select multiple vertices without making one active, it's also possible to not see a weight for any vertex.
If you want to see weights for multiple vertices, you could use this answer:
how can i use vertex weight value in shader nodes?
And in order to set weight for all selected vertices, you can do so in Object Data Properties 
Of course there's also Weight Paint Mode.
