During the medieval period in Europe (which I believe is roughly what the Witcher series is based on, along with 90%+ of modern fantasy worlds), people definitely did identify themselves (and were identified by others) based on their hometown/home region/home country/home castle. To pick some famous medieval examples of toponyms (names based on places) off the top of my head: Bernard of Angers, Henry of London or Matthew Paris.

It's not as simple as it looks however, as you might have guessed from that last one. While the famous 13th century English chronicler wrote his name as Matthaeus Parisiensis (literally 'Matthew of Paris') in Latin, he wasn't actually from Paris and the name seems to be used like a surname (as it is by modern historians e.g. 'Paris wrote his chronicle'). From around the end of the 12th century, surnames begin to develop and be passed down through families, and these sort of designations are much less reliable for pinpointing someone's place of birth or home (which causes problems for historians who try to use these names to trace medieval migration patterns). The popularity of this method of identification can be seen in the high numbers of surnames which today still have obviously refer to a place: Ireland, France, English, Fleming, Walsh (Welsh), Carlyle, Crewe, Blackburn etc.

However, toponyms were never the only way to identify oneself during the central middle ages - one could be identified by occupation (Bernard the Priest, Phillip the Doctor), by familial relationship (John son of Robert, Richard brother of Henry) or even sometimes by a nickname or distinguishing features (Ralph the Bald, William the Bastard). It's highly likely that people adapted how they identified themselves based on the context: in their home village they might be Robert son of Richard, to distinguish themselves from Robert son of Hugh who lived next door, while in the nearest town he'd be known as Robert the Blacksmith - if then went off on Crusade in a multi-ethnic army, he might be known as Robert the Norman.

TL;DR: Yes, in the medieval period people were known by toponyms, which eventually evolved into surnames from the beginning of the 13th century. However, they weren't the only way people were identified, nor even consistent in their use.