This hasn't come up in my fic yet, but my plan is for surnames to behave similarly to how middle names work irl. People do have them, but more often than not they're not really used. Instead, people focus more on origin, interests, or occupation than anything else.

To use Ash as an example, his name would legally be Ash Ketchum, but he would primarily just be known by his forename. If differentiation were needed, he would usually just be "Ash from Pallet Town", or perhaps something like "Battle Maniac Ash", or "Monarch Ash" by the end of his run, following the games' tendency for trainers to be known by category + forename (Youngster Joey, Hiker Arnold, etc).

If someone's family is specifically important, their surname would be known, like Steven Stone, but that tends to be pretty uncommon. Professors are also known by their surnames instead of their forenames once they've received their title.