Although wiring diagrams already assume associativity, I think there's a very informal way to force the concept to appear. If we want to show
\[ (x \otimes y) \otimes z = x \otimes (y \otimes z), \]
we could use positional grouping to represent the idea.
x y z
is equal to
x y z
Associativity lets us slide the wires around, while commutativity lets us slide a wire past another wire. The first line informally suggests that you have to send \\(x\\) and \\(y\\) into a box together, but associativity lets you slide the variables into different bundles and send \\(y\\) and \\(z\\) into a box instead.
To make this formal would involve cluttering up the diagram as you folks were discussing above, but the idea that associativity has to do with spacing in the diagram seems to give a partial intuition.