A plane has only 4 vertices. It is an object that cannot exist in the real world as it only has two dimensions (in this case in X and Y) It has no depth on the Z axis.
On the Z axis it has a value of 0. Multiplying 0 by any number will always yield 0, so scaling will have no effect.
Blender does not behave like sketchup, or other software, that will create new geometry just by scaling on one axis.
The way to get thickness out of a plane is to Extrude the geometry:
Select your plane, enter Edit Mode (press Tab)
Select the face (or vertices, or edges) press E and move the cursor on the axis you want the newly extruded geometry to go.
If you exit to Object mode again, you will see that your object has now a value for the dimensions in the Z axis.