Once again we have an airline disaster story short on facts. 
I would guess there's confusion between "paid for a seat" ie ticket  and "paid for a seat assignment" as in seat fee. 
Assuming the 18 yr old was booked on the flight and a seat fee was paid.
But he traveled on a different flight and we assume paid the necessary fees to change flights.
That's fine.
But when you do switch flights by your own choice the seat fee you pay is non refundable. 
The seat fee is only valid for that person on that flight in that seat.
That original seat may now be empty - assuming Delta didn't resell it- but you have no right to use it. 
Most major airlines have this policy and it's usually clearly spelled out.