Nowadays I don't give my real name out except for financial and govt related apps and services - banks, cards income tax portal etc. Most often it'll be something random or for apps that needs my name to provide me services or to communicate, it'll either be an intial or something like "Jay"If your payment details weren't saved there and you didn't reuse that password at other sites, nothing else you can do.
Unless you are open to changing your name
Of course this does not help much when they leak your address and GPS coordinates as well in a data breach, but feels a little less violated.