Why does the parent connection matter during a NEAT crossover when both connections are identical?
The genes shown in the diagrams have been simplified so that they only show topology and innovation index number. In addition, the full genome contains weights for each connection and bias for each node. 

The innovation numbers are not tracking changes with connection weights or biases, just the rough time of inserting a specific connection between two nodes. So each parent can have matching connections/innovations, but with different weights associated. The full genome is still subject to cross-over and mutation, the excerpt you quote helps explain how the genome is aligned between parents with different architectures, so that GA crossover and mutation can occur in each connection and node.

Which parent is used to create each offspring gene matters because of the additional connection weight data associated with each parent's gene.