IIRC, the issue with “Martin Guerre” was that his wife wanted a husband, so was willing to lie even though she knew he was lying; I think she got convicted along with him.
I was at a SciFi convention where L E Moddessit was giving a talk about his faux-medieval fantasy world. He was an economist and gave some interesting details. You didn’t travel with a magic American Express card, like so many characters in fiction (historical or fantasy) seem to. You needed to pay for your lodgings while travelling. Therefore, you needed to travel with a decent purse. So to avoid robbery, you needed a soldier escort… who needed to be paid, plus thier room and board at assorted Inns. So that purse got bigger. This is why in the real world, nobles travelled with a large retinue. You had to be a big enough force to deter highwaymen and gangs who would be after the small treasure you had to travel with. Not to mention the chest full of nice clothes, also a magnet for robbery.
Normally someone would travel to a known acquaintance - say the duke of Essex travelling to visit, say, the pope would carry a set of letters of introduction for others along the way, sealed by someone who either knew those hosts or from the king or someone important. A letter to the King in Paris, some Duke or Archbishop in Avignon, the duke of Milan, etc. You stayed at the host’s palace at their indulgence, and better not abuse it.
Or they would go by ship. This is the important point. People went everywhere, especially sailors and the merchants who accompanied the cargo. Marco Polo (and his uncles) were not unique, they just happened to write things down. (I think it was Ancient Engineers de Camp mentions the only written evidence we have of Roman trade with India was someone’s off-hand remark how long a ship voyage took) In those days, when connections were everything, people were a lot more concerned about who was who and what connections were there, and what were the kingdoms. The idea of showing up in London claiming to be the Duke or Laird of Inverness would be laughable.
Another important point is accent. There’s the joke about telling where in England, or even in London, someone came from by their accent. And being able to tell the details of who’s who back home to fill in their curiosity. Not to mention manners, religious observations, etc. Nothing likely pissed off the local archbishop than alleging to be a certain religion, and not knowing the details.
All in all, in a place where people had nothing better to do than gossip, one would have to be a very good imposter. You might get away with actually being from that remote place, but being a local and trying to pull it off would be unlikely. Plus, the risk of being recognized, society - especially noble society -was not that big.