Thats the symptom of a mesh not having an armature modifier set for the armature via Ctrl-P or Properties > Modifiers. Did you happen to notice if the armature had jumped back into object mode when you selected it? Probably so by your description “instead the entire skeleton is selected”.
…it’s a confusing process that I know throws off many new users, more so than other basic aspects of Blender and it does so unnecessarily.
There is nothing basic about rigging/skinning. Missing one little step in a Blender tutorial is more often than not a complete show-stopper. Poorly explained features and settings by many tutorial authors also make it difficult for new users to troubleshoot problems. You may as well get used to the fact that when first starting out you will often have to step through a tutorial multiple times before you get it right.
Welcome to BA Forums.
-Larry