In pose mode, set a Child of bone constraint on your bone, and use your object as target.

You can make a Vertex Parent:

Place an empty. In Edit mode select your vertex, Shift+S > "cursor to selected". Back to object mode, add an Empty.

Select the empty and the mesh in that order. With the empty selected, Shift+RMB on the mesh.

Make vertex parent. Back in edit mode, select the vertex, Ctrl+P > "make vertex parent".

^{the gif shows the last two steps}

If you want a bone to have a parent bone you have to do the parenting while in 'Edit Mode'. At the minute you are in 'Object Mode'. From the header of the 3D view, with the armature selected, choose 'Edit Mode':

(The bone you want to be the parent will also have to be in the same armature for it to be available as the parent).

You can select the bone you want parented, go to Properties > Bone settings > relations > Parent, and set the bone you want as a parent there.

If you have assigned your bones correctly on your character, you should be able to add additional game objects into the joints of the character, ie. Swords, shields and other cool things.

Then you can use this code to quickly fetch specific defined bones:

_playerAnimator.GetBoneTransform(HumanBodyBones.LeftHand);

Which gets the bone transform.

Here is a quick snippet of the assigned bones using a PlayerAvatar on character.