I can’t access the video right now to check, but if you want to merge vertices as in actually merging, you need to be doing that within the same mesh object. So if you create two cubes in Object mode, they will have their own independent meshes. If you create one in Object mode and the other in Edit mode for the first, then they will be of the same object mesh (even if separate).
If you are simply trying to Snap vertices together, you’ll need to enable the Snap function and set it to closest vertex in the drop-menu options.
If you are having an issue with actually just selecting a vertex that is hidden by another object, then use either C (Circle Select) or B (Box Select) while in wireframe mode to easier select things without having to turn the camera around the object.
Sometimes by trying to select a different Vertex/Edge/Face sharing the same space as another, you’ll just end up deselecting the original one. It’s tricky to accurately click when things share the same space.
Hope that helps~