If the flight has a Delta code e.g. DL2600 and is listed as R then it is First class on Delta. R is the domestic award bucket for First. I am presuming it's a KL/AF flight number and hence a KL/AF low coach fare class. The Delta app only reads R and thinks First.
Same thing happens the other way around. I have a Delta domestic flight in R and then an AF TATL flight in O. airfrance lists the domestic Delta segment as 'coach' because R on AF is a coach booking class.
This is actually very commen, on united it shows me in First on Aegan (and they don't even have First) because it's booked in O class which would be First on United.