Through out the republican and imperial periods of Roman history, important figures like consuls, generals, senators, and later emperors were issued with official types of their portraiture. This consisted of an approved (by some authority, whether the artist or subject or both is contested) bust of their head. So all statues of them would use replicas of their head to have a standard look that people could recognise. The Prima Porta type is the most famous for the emperor Augustus(I was just looking at the eponymous statue of the type yesterday in Oxford's Ashmoleon for a paper I'm writing) and would probably be recognisable to people familiar with Augustus as it has over 140 replicas still in existence. The types could change over time and are not seen as perfect examples of what they looked like although in most cases we'll never know, but we are sure of the level at which this affected the mass conception of how people thought their rulers looked. Similar forms were also used on coinage that was minted by senators and emperors which were far more widespread than any statue and handed over far and wide.

After Rome fell the level of recognisability fell greatly as the level of centralisation required for mass produced images simply was was not reached again for centuries in the west. In the east imperial images of emperors and the like was a bit more centralised but I am unaware of the specifics and how widespread images were as I am not an expert in that field unfortunately.

Hope I helped, keep learning

Correct me if I'm wrong, but wasn't Louis XVI caught during the flight to Varennes when someone actually recognized him when he went to pay for something because his face was on an assignat?

In the high to late middle ages at least, how important people dressed seems to have been how people could tell who was who. This starts on a bigger scale-- entire people groups could be told apart by how they dressed. Peasants wore, well, what you tend to picture when you think of a medieval peasant, and as you went further up the social ladder, people wore more expensive garments made with pricier exotic dyes like reds and purples, while the clergy had vestiments to show who they were, monks wore their hair tonsured, etc. In fact, in the late middle ages when wages were spiking and more people could afford more luxury goods, middle classmen were dressing like the nobility, and that became such a problem that they outlawed dressing outside your social stratum. These were called sumptuary laws, if you want to do more research there.

When it comes to individuals, yes, it's more difficult. Public appearances for lords and kings were important, but in the relatively small communities medieval cities were (compared to today), it wouldn't take much thinking to recognize, say, the local lord or the king. Individual knights were an interesting case. Literature suggests that wealthy/famous knights had very distinct armor to identify themselves by. In Malory's Morte d'artur, Lancelot steals Kay's armor and impersonates him for a while, before people recognize his fighting style and know it couldn't be anyone but Lancelot. A coat of arms was also essential, because heraldry was propaganda. Individuals would have a private coat of arms, so if you saw a certain pattern and had a good eye, you could spot Henry Bolingbroke all the way across a tourney field, for example. Every now and then we see a manuscript illumination with figures clearly depicted with consistency, but this would have been more for the sake of long-term prestige, getting immortalized in text, because the general public didn't have general access to those posh manuscripts.