The animations view doesn't have focus. Other views, like the Tree, have their name underlined when focused to indicate that is where keyboard shortcuts will take place. We could give the animations view focus, though I'm not sure it would be used for much other than duplicate.
Throughout Spine we provide a shortcut to select things by right clicking them. You can right click an animation in the animations view to select it in the tree, where you then have all the functionality like duplicate and delete. This is better than scrolling the tree to find the animation, though it isn't terribly discoverable.
Right click works in other places, like the preview and skins view.
You can also click the animation name in the first row of the dopesheet or graph, making for an easy way to reach it that is almost always visible.