The tense situation dives deep into some unclear rules and recommendations when another person who is presumably a Delta employee appears and cites a FAA regulation. "With him being two, he cannot sit in the car seat," she says. That's not true. In fact, despite any perceived murkiness in the rules, it's almost certainly the opposite.
Children over the age of two are required to have their own seat. Maybe the employee was rounding up, but even if the child is under two, the employee is still incorrect. Delta's rules state a child under two "may" sit on the lap of an adult. They recommend children sit in a child's seat. Additionally, the FAA "strongly urges" parents put children in a safety seat and FAA rules say an airline cannot prevent an adult from putting a child in such a seat.
The long argument highlights the absurdity of rigid no-transfer rules. (Technically, children under 18 don't need to show ID. The airline would not be able to prove it was the wrong child in the seat if the family doesn't disclose that information.) Eventually, the family says, for the sake of taking off, they'll put their son on a lap. "It's come too far," the family is told. "At this point, you guys are on your own."