The flower named the color
Dianthus means flower of the gods, from the Greek. The English name pink refers to the petal edge, which looks cut with pinking shears, and the verb is the older word. The color took its name from the flower in the seventeenth century, not the other way around.
Carnation is a separate puzzle nobody has settled. Coronation is the most likely origin, from the flower's use in garlands, though incarnation and a reference to flesh color both have their supporters.
The clove scent in an old fashioned border pink is not a coincidence either. These species carry eugenol, the compound that gives cloves their smell, which is why the flower was once used to spice wine. Most modern commercial carnations have had that scent bred almost entirely out in favor of size and vase life. It is a real loss, and worth mentioning to any customer old enough to remember it.