I think you misunderstand what the bottom box is for.
It's a property box for the thing that's currently selected in the tree at the time indicated in the dopesheet (where the teal line is).
It's not the property box of keys in the dopesheet. Applying the key only applies it to whatever you selected in the tree. It doesn't matter what you select in the dopesheet.
Strangely, if you have the dopesheet locked (with the tiny padlock button), when you select on a key, it will select that item in the tree. That may lead to the workflow you're looking for.
But if you select two keys of two different items, those items will not be multi-selected in the tree.
