At the risk of pouring fuel on a smoldering fire, I don't believe the use of saline is anything to do with birds vs mammals. It is actually basic physiology - related to osmosis (which is the passive movement of water across cell membranes).
Plain water is hypotonic relative to any living cell. Any cell placed in a hypotonic solution will swell as water moves into the cell.
Strongly salty water is hypertonic and any cell placed in that solution will lose water and shrink. It is why if you soak a wound or boil in strongly salty water the swelling will reduce. It also stings like crazy so a better home remedy is to make a hypertonic solution using old bar soap.
If you have ever been given IV fluids in a hospital you have received something called 'Normal Saline' (which is 9% saline). This is sterile salty water that is isotonic to human blood. It is used to resuscitate people not because they need the salt but because if you used plain water then your blood cells would burst.
So, if you are trying to clean an eye - any species - then either plain water or salty water is just fine (as long as it is sterile). Plain water will not help any swelling and some will worry that the cells on the surface of the eye might swell, salty water might reduce swelling in cells on the surface of the eye, but it is probably a marginal impact.
I will shut up now and let everyone argue about vets and salty water - you all just triggered my inner physiology geek.
Apart from my old text books which will take too long to dig up I give you a basic reference from the NIH.
https://www.ncbi.nlm.nih.gov/books/NBK557609/