I'm trying to Box Select some vertices but for some reason it has problems selecting overlapping vertices.
Here's a video that demonstrates this behaviour: https://streamable.com/hvj34
What am I doing wrong here?
The selection tools by default only select vertices not hidden by other faces. In your case, the vertices very close to each other (on on the same point) are preventing one or other of each pair from being selected.
To avoid this ‘masking’ you can swap to wireframe mode (Z) before making the selection. This will allow you to select all vertices within the box region.