The vertical height isn't relevant, but the relative vertical height between two "nodes" on different wires could (a-priori) be. The fact that you can reorder them as you like so only the topology matters is exactly encoding the fact that they are natural transformations, if they were some other map it wouldn't work.
To say it explicitly, string diagrams are nice exactly because they convert category theoretic equivalences into topological equivalences that are easy to read visually.
String diagrams are also pretty confusing if you're not already very comfortable with category theory. They exist to make complex equality proofs a little easier, but the underlying proofs themselves are not easy to follow.
