Cougar is apparently his middle name. Maybe just for stage purposes.
Johnny Cougar was a stage name. He tells a story of his grandfather never voting after someone at the polls made fun of the name “Mellencamp.” After he made it he changed to John Cougar Mellencamp.
His name is John Mellencamp. The his manager and the suits in the record company thought it wasn’t marketable and made him take the stage name of Johnny Cougar and later John Cougar. He hated it. But he had a couple of minor hits under the name so he was sort of stuck. But with hits comes some clout and he wanted to be himself so he took slow steps away from Johnny Cougar, first changing it to John Cougar Mellencamp and then dropping the Cougar altogether.
Cougar was a stage name.