The World Z-Axis is up in Blender, that is just a (more or less) arbitrary decision made one time. There is other Software that uses the Y-Axis up for the World.
An Armature Object, in its default Orientation (as it comes in, when you Add an Armature) is aligned to the World and thus has its Z-Axis up.
A Bone has its Y-Axis along the length of the Bone (from Head to Tail). (I don’t know if this is universal, or Software dependent.)
A Mesh Object comes by default in, aligned to the World, so with its Z-Axis up.
A Cube, for instance, (in Edit Mode) has 6 Faces and each Face has its Z-Axis in a different orientation.
Please don’t overthink this.
