You link proves my point. I have never needed any ID for my child except a passport when we are traveling out of the country.
Aren’t you special.
The link says that airlines may require something like a copy of a birth certificate to confirm age and that parents should have such with them.
If a passenger is looking to switch to a new passenger name for a child traveling with them, it is reasonable for the airline to ask and maybe even confirm with documentation such child’s age.
That, as I said originally, is where IMO the airline should have caught this. And that would have been the time and place, if any, for them to disallow the parents giving a small child a seat against the regular policy.