Yes, the statues were indeed usually named. The portraits which populated the public spaces of Greco-Roman cities were normally part of honorific monuments that also included some kind of base with an inscription providing the name of the person and the reasons why they were honoured. A rather late (around 500 AD) but also very well preserved example comes from the Carian city of Aphrodisias in modern Turkey. Without the accompanying inscription we would have no way of knowing that this guy’s name was Flavius Palmatus and that he served as Roman governor in western Asia Minor. So ancient onlookers should have had no problem to identify the various public portraits they were confronted with, at least as long as they could read or had someone to read the inscriptions for them. The problem here is of course that the vast majority of ancient statue monuments aren’t nearly as well preserved as the above example from Aphrodisias. Take for instance another Late Roman head from Asia Minor. It was found at Ephesus in the proximity of an inscription honouring a man called Eutropius for financing the pavement of a nearby street with marble. Therefor the head is usually identified with this man, but in this case we can’t be nearly as sure as with Palmatus. For the majority of ancient portraits there is no inscription at all to help us. Of course, that hasn’t stopped researchers from trying to identify them anyway. Seems to be part of human nature to want to put a specific name to individual faces. Unfortunately those identifications are often based on rather flimsy grounds.

So, what about Roman emperors? Do we have extant statue-inscription-combos for every one of them? There were far more portraits of them to be found all over the empire than of any other individuals so the chances to find such a well-preserved monument are at least much higher than for anybody else. Nevertheless, I doubt there is one for every single emperor. Thankfully, this isn’t necessary anyway. There is another medium in which their likenesses were routinely combined with inscriptions of their names and titles: coinage. Every new ruler put his own portrait on the coins put out during his rule and we have far more extant examples for those than for honorific statues. Also, these imperial portraits were highly standardised even down to tiny details, so it’s relatively easy to compare and identify them. Let’s make such a comparison with some examples from the reign of Augustus. The facial features on the coin and the statue are very similar and even the individual locks seem to be reproduced. That later fact becomes most obvious once we look at the pair of locks above the nose and the left eye. In the frontal picture of the statue we see them forming a kind of pincer. On the coin that form is reproduced directly above the forehead. Thanks to these kinds of details we can identify the portraits of pretty much every emperor of the Early and High Imperial Age. It’s only in Late Antiquity that imperial portraits become far less individualised and much harder to connect to any specific ruler. For example, the so called ‘colossus of Barletta’, a monumental Late Roman bronze statue, has variously identified with emperors like Valentinian I, Valens, Markian or Leo I. But if you’re googling for pictures of Marcus Aurelius you can be reasonably confident that the search results will really show you the author of the ‘Meditations’ and not anybody else.