The connection between Rinoa and Ultimecia is part of the plot.

Rinoa has White Wings. And if you look at her back, it looks like the wings of a bird.
Ultimecia has Black Wings. And if you look at her back, it looks like the wings of a bird.

Yet, you see living white birds and black birds in this tunnel. The lightning being struck was what triggered the black birds to change into white birds.

Therefore, they are all connected. The birds are here to represent something.