What does a person, who has a waiver visa (ESTA) to visit USA, need to fly within USA?
What information have to be filled during flight booking during purchase (domestic flights usually don't ask for passport number)?
You need to bring your passport, which is the only TSA-approved foreign ID for flying in the US.
And that's it, really. On booking, you'll be asked for name, age and contact details, but none of this is checked or verified. At the airport, all they'll do is verify that the name on your ticket matches the name on your ID.